The Nonlinear Schr¨odinger Equation, Superﬂuidity and ...bao/PS/fudan.pdf · The Nonlinear...

The Nonlinear Schrodinger Equation, Superfluidity and

Quantum Hydrodynamics

Weizhu Bao

Department of Computational ScienceNational University of Singapore, Singapore 117543Email: [email protected] Fax: 65-67746756

URL: http://www.cz3.nus.edu.sg/˜bao/

Contents

1 The nonlinear Schrodinger equation 3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Derivation of NLSE from wave propagation . . . . . . . . . . . . . . . . . . 4

1.3 Derivation of NLSE from BEC . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.1 Dimensionless GPE . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.2 Reduction to lower dimension . . . . . . . . . . . . . . . . . . . . . . 7

1.4 The NLSE and variational formulation . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.2 Lagrangian structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.3 Hamiltonian structure . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.4 Variance identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Plane and solitary wave solutions . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6 Existence/blowup results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6.1 Integral form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6.2 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6.3 Finite time blowup results . . . . . . . . . . . . . . . . . . . . . . . . 17

1.7 WKB expansion and quantum hydrodynamics . . . . . . . . . . . . . . . . . 17

1.8 Wigner transform and semiclassical limit . . . . . . . . . . . . . . . . . . . . 18

1.9 Ground, excited and central vortex states . . . . . . . . . . . . . . . . . . . 20

1.9.1 Stationary states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.9.2 Ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.9.3 Central vortex states . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.9.4 Variation of stationary states over the unit sphere . . . . . . . . . . 23

1.9.5 Conservation of angular momentum expectation . . . . . . . . . . . 24

1.10 Numerical methods for computing ground states . . . . . . . . . . . . . . . 251.10.1 Gradient flow with discrete normalization (GFDN) . . . . . . . . . . 25

1.10.2 Energy diminishing of GFDN . . . . . . . . . . . . . . . . . . . . . . 26

1.10.3 Continuous normalized gradient flow (CNGF) . . . . . . . . . . . . . 27

1.10.4 Semi-implicit time discretization . . . . . . . . . . . . . . . . . . . . 28

1.10.5 Discretized normalized gradient flow (DNGF) . . . . . . . . . . . . . 301.10.6 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.10.7 Energy diminishing of DNGF . . . . . . . . . . . . . . . . . . . . . . 33

1.10.8 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.11 Numerical methods for dynamics of NLSE . . . . . . . . . . . . . . . . . . . 40

1.11.1 General high-order split-step method . . . . . . . . . . . . . . . . . . 40

1.11.2 Fourth-order TSSP for GPE without external driven field . . . . . . 41

1

2 CONTENTS

1.11.3 Second-order TSSP for GPE with external driven field . . . . . . . . 421.11.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.11.5 Crank-Nicolson finite difference method (CNFD) . . . . . . . . . . . 441.11.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2 The Zakharov system 512.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.2 Derivation of the vector Zakharov system . . . . . . . . . . . . . . . . . . . 512.3 Generalization and simplification . . . . . . . . . . . . . . . . . . . . . . . . 55

2.3.1 Reduction from VZSM to GVZS . . . . . . . . . . . . . . . . . . . . 562.3.2 Reduction from GVZS to GZS . . . . . . . . . . . . . . . . . . . . . 572.3.3 Reduction from GVZS to VNLS . . . . . . . . . . . . . . . . . . . . 592.3.4 Reduction from GZS to NLS . . . . . . . . . . . . . . . . . . . . . . 602.3.5 Add a linear damping term to arrest blowup . . . . . . . . . . . . . 60

2.4 Well-posedness of ZS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.5 Plane wave and soliton wave solutions . . . . . . . . . . . . . . . . . . . . . 612.6 Time-splitting spectral method . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.6.1 Crank-Nicolson leap-frog time-splitting spectral discretizations (CN-LF-TSSP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.6.2 Phase space analytical solver + time-splitting spectral discretiza-tions (PSAS-TSSP) . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.6.3 Properties of the numerical methods . . . . . . . . . . . . . . . . . . 682.6.4 Extension TSSP to GVZS . . . . . . . . . . . . . . . . . . . . . . . . 70

2.7 Crank-Nicolson finite difference (CNFD) method . . . . . . . . . . . . . . . 732.8 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3 The Maxwell-Dirac system 793.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.2 The Maxwell-Dirac system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.2.1 Dimensionless Maxwell-Dirac system . . . . . . . . . . . . . . . . . . 813.2.2 Plane wave solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.3 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.3.1 Time-splitting spectral discretization . . . . . . . . . . . . . . . . . . 843.3.2 Properties of the numerical method . . . . . . . . . . . . . . . . . . 923.3.3 For homogeneous Dirichlet boundary conditions . . . . . . . . . . . . 95

3.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Chapter 1

The nonlinear Schrodingerequation

1.1 Introduction

The Schrodinger equation was proposed to model a system when the quantum effect wasconsidered. For a system with N atoms, the Schrodinger equation is defined in 3N + 1dimensions. With such high dimensions, even use today’s supercomputer, it is impossibleto solve the Schrodinger equation for dynamics of N atoms with N > 10. After assumedHatree-Fork ansatz, the 3N +1 dimensions linear Schrodinger equation was approximatedby a 3+1 dimensions nonlinear Schrodinger equation (NLSE) or Schrodinger-Poisson (S-P)system. Although nonlinearity in NLSE brought some new difficulties, but the dimensionswere reduced significantly compared with the original problem. This opened a light tostudy dynamics of N atoms when N is large. Later, it was found that NLSE had applica-tions in different subjects, e.g. quantum mechanics, solid state physics, condensed matterphysics, quantum chemistry, nonlinear optics, wave propagation, optical communication,protein folding and bending, semiconductor industry, laser propagation, nano technologyand industry, biology etc. Currently, the study of NLSE including analysis, numerics andapplications becomes a very important subject in applied and computational mathemat-ics. This study has very important impact to the progress of other science and technologysubjects.

In this chapter, we first review derivation of NLSE from wave propagation and Bose-Einstein condensation (BEC). Then we present variational formulation of NLSE includ-ing conservation laws, Lagrangian structure, Hamiltonian structure and variance identity.Plane and soliton wave solutions, existence/blowup results of NLSE are then presented.Ground, excited and central vortex states of NLSE with an external potential are studied.We also study formally semiclassical limits of NLSE by WKB expansion and Wigner trans-form when the (scaled) Planck constant ε→ 0. Finally numerical methods for computingground states and dynamics of NLSE are presented. Numerical results are also reported.

Throughout this notes, we use f∗, Re(f) and Im(f) denote the conjugate, real part andimaginary part of a complex function f respectively. We also adopt the standard Sobolevnorms.

3

4 CHAPTER 1. THE NONLINEAR SCHRODINGER EQUATION

1.2 Derivation of NLSE from wave propagation

In this section, we review briefly derivation of NLSE from wave propagation, i.e. parabolicor paraxial approximation for forward propagation time harmonic waves, to analyze highfrequency asymptotics.

The wave equation

1

c2∂2u(x, t)

∂t2−4u(x, t) = 0, x ∈ R3, (2.1)

where x = (x, y, z) is the Cartesian coordinate, t is time and c = c(x, |u|) is the propagationspeed, has time harmonic solutions of the form eiωtu(x) with the complex wave functionu satisfying the Helmholtz or reduced wave equation

4u(x) +ω2

c2u = 0, x ∈ R3. (2.2)

Let c0 be a uniform reference speed, k0 = ω/c0 be the wave number and n(x, |u|) =c0/c(x, |u|) be the index of refraction. The reduced wave equation has then the form

4u(x) + k20n

2(x, |u|)u = 0. (2.3)

When wave propagates in a uniform medium, n(x, |u|) = 1; in a linear medium, n(x, |u|) =n(x); and in a Kerr medium, n(x, |u|) =

√1 + 4n2|u|2/n0 with n0 linear index of refraction

and n2 Kerr coefficient.When waves are approximately plane and move in one direction primarily, say the z

direction, e.g. propagation of laser beams, we look for solutions of the form

u(x, y, z) = eik0zψ(x, z) (2.4)

where x = (x, y) denotes the transverse variables. We insert (2.4) into the reduced waveequation (2.3) and get

2ik0ψz + 4⊥ψ + k20µ(x, z, |ψ|)ψ + ψzz = 0, (2.5)

where 4⊥ is the Laplacian in the transverse variables and µ(x, z, |ψ|) = n2(x, z, |ψ|) − 1is the fluctuation in the refractive index. Note that the direction of propagation z ploysthe role of time and −k2

0µ(x, z, |ψ|) is the (time dependent) potential.Introduce nondimensional variables:

x =x

r0, y =

y

r0, t =

z

k0r20, ψ(x, y, t) =

ψ(x, y, z)

ψs, (2.6)

where r0 is the dimensionless length unit, e.g. width of the input laser beam, and ψs isdimensionless unit for ψ to be determined. Plugging (2.6) into (2.5), multiplying by r20/2,and then removing all ˜, we get the following dimensionless equation:

iψt = −1

24⊥ ψ + f(x, t, |ψ|)ψ − δ

2ψtt, (2.7)

where δ = 1/r20k20 and the real-valued function f depends on µ. Due to the input beam

width r0 λ = 2π/k0, we get

δ/2 = λ2/8π2r20 1.

1.3. DERIVATION OF NLSE FROM BEC 5

Thus we drop the nonparaxial term ψtt in (2.7) and obtain the NLSE:

iψt = −1

24⊥ ψ + f(x, t, |ψ|)ψ, (2.8)

Of course (2.7) is only an approximation to the full reduced wave equation and it isvalid when the variations in the index of refraction are smooth and the bulk of the waveenergy is away from boundaries. This important and very useful approximation for wavepropagation is well suited for numerical approximation since we now have an initial valueproblem for ψ, assuming that ψ(x, 0) is known, rather than a boundary value problem foru.

When n(x, z, |u|) = 1 in (2.3), then µ(x, z, |ψ|) = 0 in (2.5) and f(x, t, |ψ|) = 0 in(2.7), thus (2.7) collapses to the free Schrodinger equation:

iψt = −1

24⊥ ψ. (2.9)

When n(x, z, |u|) = n(x, z) in (2.3), then µ(x, z, |ψ|) = µ(x, z) in (2.5) and f(x, t, |ψ|) =V (x, t) in (2.7), thus (2.7) collapses to a linear Schrodinger equation with potential V (x, t):

iψt = −1

24⊥ ψ + V (x, t)ψ. (2.10)

when n(x, z, |u|) =√

1 + 4n2|u|2/n0, i.e. laser beam in Kerr medium, then µ(x, z, |ψ|) =2n2r

20k

20|ψ|2/n0 in (2.5) and f(x, t, |ψ|) = −|ψ|2 in (2.7) by choosing ψs =

√n0/r0k0

√2n2,

thus (2.7) collapses to NLSE with a cubic focusing nonlinearity:

iψt = −1

24⊥ ψ − |ψ|2ψ. (2.11)

The wave energy or power of the beam is conserved:

N(ψ) =

∫

R2|ψ(x, t)|2 dx ≡

∫

R2|ψ(x, 0)|2 dx, t ≥ 0. (2.12)

Remark 1.2.1 When we consider high frequency asymptotics which concerns approximatesolutions of (2.10) that are good approximations to oscillatory solutions. For such solutionsthe propagation distance is long compared to the wavelength, the propagation time is largecompared to the period and the potential V (x) varies slowly. To make this precise, weintroduce slow time and space variables t → t/ε, x → x/ε with 0 < ε 1 the (scaled)Planck constant and the scaled wave function ψε(x, t) = ψ(x/ε, t/ε) which satisfies theNLSE in the semiclassical regime

iεψεt = −ε2

24⊥ ψ

ε + V ε(x, t)ψε, x ∈ R2, t > 0, (2.13)

where V ε(x, t) = V (x/ε, t/ε).

1.3 Derivation of NLSE from BEC

Since its realization in dilute bosonic atomic gases [3, 26], BEC of alkali atoms and hy-drogen has been produced and studied extensively in the laboratory [71], and has spurred


great excitement in the atomic physics community and renewed the interest in studyingthe collective dynamics of macroscopic ensembles of atoms occupying the same one-particlequantum state [99, 34, 68]. The condensate typically consists of a few thousands to mil-lions of atoms which are confined by a trap potential. In fact, beside the effects of theinternal interactions between the atoms, the macroscopic behavior of BEC matter is highlysensitive to the shape of this external trapping potential. Theoretical predictions of theproperties of a BEC like the density profile [19], collective excitations [43] and the forma-tion of vortices [105] can now be compared with experimental data [3]. Needless to saythat this dramatic progress on the experimental front has stimulated a wave of activityon both the theoretical and the numerical front.

At temperatures T much smaller than the critical temperature Tc [84], a BEC is welldescribed by the macroscopic wave function ψ = ψ(x, t) whose evolution is governed by aself-consistent, mean field NLSE known as the Gross-Pitaevskii equation (GPE) [69, 103]

ih∂ψ(x, t)

∂t=

δH(ψ)

δψ∗

= − h2

2m∇2ψ(x, t) + V (x)ψ(x, t) +NU0|ψ(x, t)|2ψ(x, t), (3.1)

where x = (x, y, z), m is the atomic mass, h is the Planck constant, N is the number ofatoms in the condensate, V (x) is an external trapping potential. When a harmonic trap

potential is considered, V (x) = m2

(ω2xx

2 + ω2yy

2 + ω2zz

2)

with ωx, ωy and ωz being the

trap frequencies in x, y and z-direction, respectively. The Hamiltonian (or energy) of thesystem H(ψ) per particle is defined as

H(ψ) =

∫

R3ψ∗(x, t)

[− h2

2m∇2 + V (x)

]ψ(x, t)dx

+1

2

∫

R3×R3ψ∗(x, t) ψ∗(x′, t)Φ(x − x′)ψ(x′, t)ψ(x, t)dxdx′ , (3.2)

where the interaction potential is taken as the Fermi form

Φ(x) = (N − 1)U0δ(x). (3.3)

U0 = 4πh2as/m describes the interaction between atoms in the condensate with the s-wave scattering length as (positive for repulsive interaction and negative for attractiveinteraction). It is convenient to normalize the wave function by requiring

∫

R3|ψ(x, t)|2 dx = 1. (3.4)

1.3.1 Dimensionless GPE

In order to scale the Eq. (3.1) under the normalization (3.4), we introduce

t = ωmt, x =x

a0, ψ(x, t) = a

3/20 ψ(x, t), with a0 =

√h/mωm, (3.5)

where ωm = minωx, ωy, ωz, a0 is the length of the harmonic oscillator ground state.In fact, we choose 1/ωm and a0 as the dimensionless time and length units, respectively.

1.3. DERIVATION OF NLSE FROM BEC 7

Plugging (3.5) into (3.1), multiplying by 1/mω2ma

1/20 , and then removing all ˜, we get the

following dimensionless GPE under the normalization (3.4) in three dimension

i∂ψ(x, t)

∂t= −1

2∇2ψ(x, t) + V (x)ψ(x, t) + β |ψ(x, t)|2ψ(x, t), (3.6)

where β = U0Na30hωm

= 4πasNa0

and

V (x) =1

2

(γ2xx

2 + γ2yy

2 + γ2zz

2), with γα =

ωαωm

(α = x, y, z).

There are two extreme regimes of the interaction parameter β: (1) β = o(1), the Eq.(3.6) describes a weakly interacting condensation; (2) β 1, it corresponds to a stronglyinteracting condensation or to the semiclassical regime.

There are two typical extreme regimes between the trap frequencies: (1) γx = 1, γy ≈ 1and γz 1, it is a disk-shaped condensation; (2) γx = 1, γy 1 and γz 1, it is acigar-shaped condensation. In these two cases, the three-dimensional (3D) GPE (3.6) canbe approximately reduced to a 2D and 1D equation respectively [85, 8, 5] as explainedbelow.

1.3.2 Reduction to lower dimension

Case I (disk-shaped condensation):

ωx ≈ ωy, ωz ωx, ⇐⇒ γx = 1, γy ≈ 1, γz 1.

Here, the 3D GPE (3.6) can be reduced to a 2D GPE with x = (x, y) by assuming thatthe time evolution does not cause excitations along the z-axis, since the excitations alongthe z-axis have large energy (of order hωz) compared to that along the x and y-axis withenergies of order hωx. Thus, we may assume that the condensation wave function alongthe z-axis is always well described by the harmonic oscillator ground state wave function,and set

ψ = ψ2(x, y, t)φho(z) with φho(z) = (γz/π)1/4 e−γzz2/2. (3.7)

Plugging (3.7) into (3.6), multiplying by φ∗ho(z), integrating with respect to z over (−∞,∞),we get

i∂ψ2(x, t)

∂t= −1

2∇2ψ2 +

1

2

(γ2xx

2 + γ2yy

2 + C)ψ2 + β2|ψ2|2ψ2, (3.8)

where

β2 = β

∫ ∞

−∞φ4

ho(z) dz = β

√γz2π, C =

∫ ∞

−∞

(γ2zz

2|φho(z)|2 +

∣∣∣∣dφho

dz

∣∣∣∣2)dz.

Since this GPE is time-transverse invariant, we can replace ψ2 → ψ e−iCt2 so that the

constant C in the trap potential disappears, and we obtain the 2D effective GPE:

i∂ψ(x, t)

∂t= −1

2∇2ψ +

1

2

(γ2xx

2 + γ2yy

2)ψ + β2|ψ|2ψ. (3.9)

Note that the observables, e.g. the position density |ψ|2, are not affected by dropping theconstant C in (3.8).


Case II (cigar-shaped condensation):

ωy ωx, ωz ωx ⇐⇒ γx = 1, γy 1, γz 1.

Here, the 3D GPE (3.6) can be reduced to a 1D GPE with x = x. Similarly as in the 2Dcase, we can derive the following 1D GPE [85, 8, 5]:

i∂ψ(x, t)

∂t= −1

2ψxx(x, t) +

γ2xx

2

2ψ(x, t) + β1|ψ(x, t)|2ψ(x, t), (3.10)

where β1 = β√γyγz/2π.

The 3D GPE (3.6), 2D GPE (3.9) and 1D GPE (3.10) can be written in a unified form:

i∂ψ(x, t)

∂t= −1

2∇2ψ + Vd(x)ψ + βd |ψ|2ψ, x ∈ Rd, (3.11)

ψ(x, 0) = ψ0(x), x ∈ Rd, (3.12)

with

βd = β

√γyγz/2π,√γz/2π,

1,

Vd(x) =

γ2xx

2/2, d = 1,(γ2xx

2 + γ2yy

2)/2, d = 2,(

γ2xx

2 + γ2yy

2 + γ2zz

2)/2, d = 3,

(3.13)

where γx > 0, γy > 0 and γz > 0 are constants. The normalization condition for (3.11) is

N(ψ) = ‖ψ(·, t)‖2 =

∫

Rd|ψ(x, t)|2 dx ≡

∫

Rd|ψ0(x)|2 dx = 1. (3.14)

Remark 1.3.1 When βd 1, i.e. in a strongly repulsive interacting condensation or insemiclassical regime, another scaling of the GPE (3.11) is also very useful. In fact, aftera rescaling in (3.11) under the normalization (3.14): x → ε−1/2x and ψ → εd/4ψ with

ε = β−2/(d+2)d , then the GPE (3.11) can be rewritten as

iε∂ψ(x, t)

∂t= −ε

2

2∇2ψ + Vd(x)ψ + |ψ|2ψ, x ∈ Rd. (3.15)

1.4 The NLSE and variational formulation

Consider the following NLSE:

iψt = −1

24 ψ + V (x)ψ + β|ψ|2σψ, x ∈ Rd, t ≥ 0, (4.1)

ψ(x, 0) = ψ0(x), x ∈ Rd, (4.2)

where σ > 0 is a positive constant (σ = 1 corresponds to a cubic nonlinearity and σ = 2corresponds to a quintic nonlinearity), V (x) is a real-valued potential whose shape isdetermined by the type of system under investigation, β positive/negative corresponds todefocusing/focusing NLSE.

1.4. THE NLSE AND VARIATIONAL FORMULATION 9

1.4.1 Conservation laws

Two important invariants of (4.1) are the normalization of the wave function

N(ψ(·, t)) =

∫

Rd|ψ(x, t)|2 dx ≡ N = N(ψ0), t ≥ 0 (4.3)

and the energy

E(ψ(·, t)) =

∫

Rd

[1

2|∇ψ(x, t)|2 + V (x)|ψ(x, t)|2 +

β

σ + 1|ψ(x, t)|2σ+2

]dx

= E(ψ0), t ≥ 0. (4.4)

When V (x) ≡ 0, another important invariant of (4.1) is the momentum

P(ψ(·, t)) =i

2

∫

Rd(ψ∇ψ∗ − ψ∗∇ψ) dx ≡ P(ψ0), t ≥ 0. (4.5)

Define the mass center

x(t) =1

N

∫

Rdx|ψ(x, t)|2 dx. (4.6)

Note that the mass center obeys

Ndx

dt=

∫x∂t|ψ|2 dx = − i

2

∫x∇ · [ψ∇ψ∗ − ψ∗ 4 ψ] dx

=i

2

∫[ψ∇ψ∗ − ψ∗ 4 ψ] dx = P(ψ0) (4.7)

and thus moves at a constant speed.To get more conservation laws, one can use the Noether theorem [113].

1.4.2 Lagrangian structure

Define the Lagrangian density L associated to (4.1) in terms of the real and imaginaryparts u and v of ψ, or equivalently in terms of ψ and ψ∗ viewed as independent variablesin the form

L =i

2(ψ∗ψt − ψψ∗

t ) −1

2∇ψ · ∇ψ∗ − V (x)ψψ∗ − β

σ + 1ψσ+1(ψ∗)σ+1. (4.8)

Consider the action

Sψ,ψ∗ =

∫ t1

t0

∫

RdL dxdt (4.9)

as a functional on all admissible regular function satisfying the prescribed conditionsψ(x, t0) = ψ0(x) and ψ(x, t1) = ψ1(x). Its variation

δS = Sψ + δψ, ψ∗ + δψ∗ − Sψ,ψ∗ (4.10)

for infinitesimal δψ and δψ∗ reads

δS =

∫ t1

t0

∫

Rd

[∂L∂ψ

δψ +∂L∂∇ψ · ∇δψ +

∂L∂ψt

δψt

]dxdt + c.c.

=

∫ t1

t0

∫

Rd

[∂L∂ψ

−∇ ·(∂L∂∇ψ

)− ∂t

(∂L∂ψt

)]δψ dxdt

+

[∂L∂ψt

δψ

]t1

t0

+ c.c. (4.11)


A necessary and sufficient condition for a function ψ(x, t) to lead to an extremum forthe action S among the functions with prescribed values ψ(·, t0) and ψ(·, t1), thus reducesto the Euler-Lagrange equations

∂L∂ψ

= ∇ ·(∂L∂∇ψ

)+ ∂t

(∂L∂ψt

), (4.12)

which, when the Lagrangian (4.8) is used, reduces to the NLSE (4.1). This system is easilyrewritten in terms of the real fields u = (ψ + ψ∗)/2 and v = (ψ − ψ∗)/2i as

∂L∂u

= ∇ ·(∂L∂∇u

)+ ∂t

(∂L∂ut

), (4.13)

∂L∂v

= ∇ ·(∂L∂∇v

)+ ∂t

(∂L∂vt

). (4.14)

1.4.3 Hamiltonian structure

As usual, a Hamiltonian structure is easily derived from the existence of a Lagrangian.Writing ψ = u+iv in order to deal with real fields, the Hamiltonian density H = i

2(ψ∗∂tψ−ψ∂tψ

∗) − L becomes

H = v∂tu− u∂tv − L. (4.15)

Introducing the canonical variables

q1 ≡ u, p1 ≡ ∂L∂(∂tq1)

, (4.16)

q2 ≡ v, p2 ≡ ∂L∂(∂tq2)

, (4.17)

it takes the form

H =∑

j

pj∂tqj − L. (4.18)

Define

ρj ≡∂L

∂(∇qj), (4.19)

and rewrite the Euler-Lagrange equations as

∂L∂q

= ∇ · ρj + ∂tpj. (4.20)

Using that

∂tL =∑

j

∂L∂qj

∂tqj +∂L∂∇qj

∂t∇qj +∂L

∂(∂tqj)∂ttqj , (4.21)

and the Euler-Lagrange equations, we get

∂tH = −∇ ·∑

j

ρj∂tqj, (4.22)

which ensures that the conservation fo the Hamiltonian or energy H =∫Rd H dx.


Similarly, from the variation of the Lagrangian density

δL =∑

j

δLδqj

δqj +δLδ∇qj

∇δqj +δL

δ(∂tqj)∂tδqj , (4.23)

the Euler-Lagrange equations, and the definition of pj we obtain the variation of theHamiltonian H, in the form

δH =∑

j

∫(∂tqjδpj − ∂tpjδqj)dx, (4.24)

which leads to the Hamilton equaitons

∂qj∂t

=δH

δpj,

∂pj∂t

= −δHδqj

, (4.25)

or in complex form,

i∂tψ =δH

δψ∗. (4.26)

1.4.4 Variance identity

Define the ‘variance’ (or ‘momentum of inertia’ in a context where N is referred to as themass of the wave packet) as

δV

=

∫

Rd|x|2|ψ|2 dx =

d∑

j=1

δj , δj =

∫

Rdx2j |ψ|2 dx, j = 1, · · · , d (4.27)

and the square width of the wave packet

δx =1

N

∫

Rd|x− x|2|ψ|2 dx =

1

N

∫

Rd(|x|2 − |x|2)|ψ|2 dx =

δV

N− |x|2. (4.28)

Here we use x = (x1, · · · , xd) ∈ Rd.

When V (x) ≡ 0 in (4.1), due to the conservation of the wave energy N and of thelinear momentum P, we have

d2δxdt2

=1

N

d2δV

dt2− 2

|P|2N2

. (4.29)

Lemma 1.4.1 Suppose ψ(x, t) be the solution of the problem (4.1), (4.2), then we have

d2δj(t)

dt2=

∫

Rd

[2|∂xjψ|2 +

2σβ

σ + 1|ψ|2σ+2 − 2xj |ψ|2∂xj (V (x))

]dx, t ≥ 0, (4.30)

δj(0) = δ(0)j =

∫

Rdx2j |ψ0(x)|2 dx, j = 1, · · · , d, (4.31)

δ′j(0) = δ(1)j = 2 Im

[∫

Rdxjψ

∗0 ∂xjψ0dx

]. (4.32)


Proof: Differentiate (4.27) with respect to t, notice (4.1), integrate by parts, we have

dδj(t)

dt=

d

dt

∫

Rdx2j |ψ(x, t)|2 dx =

∫

Rdx2j (ψ ∂tψ

∗ + ψ∗ ∂tψ) dx

=i

2

∫

Rdx2j (ψ∗ 4 ψ − ψ4 ψ∗) dx = i

∫

Rdxj(ψ ∂xjψ

∗ − ψ∗ ∂xjψ)dx. (4.33)

Similarly, differentiate (4.33) with respect to t, notice (4.1), integrate by parts, we have

d2δj(t)

dt2= i

∫

Rdxj[∂tψ ∂xjψ

∗ + ψ ∂xj tψ∗ − ∂tψ

∗ ∂xjψ − ψ∗ ∂xj tψ]dx

=

∫

Rd

[2ixj

(∂tψ ∂xjψ

∗ − ∂tψ∗ ∂xjψ

)+ i (ψ∗ ∂tψ − ψ ∂tψ

∗)]dx

=

∫

Rd

[−xj

(∂xjψ

∗ 4 ψ + ∂xjψ 4 ψ∗)

+ 2xjV (x)(ψ ∂xjψ

∗ + ψ∗ ∂xjψ)

+2βxj |ψ|2σ(ψ ∂xjψ

∗ + ψ∗ ∂xjψ)− 1

2(ψ∗ 4 ψ + ψ 4 ψ∗)

+2V (x)|ψ|2 + 2β|ψ|2σ+2]dx

=

∫

Rd

[2|∂xjψ|2 − |∇ψ|2 − |ψ|2∂xj (2xjV (x)) + |∇ψ|2 − 2β

σ + 1|ψ|2σ+2

+2V (x)|ψ|2 + 2β|ψ|2σ+2]dx

=

∫

Rd

[2|∂xjψ|2 +

2σβ

σ + 1|ψ|2σ+2 − 2xj |ψ|2 ∂xj (V (x))

]dx. (4.34)

Thus we obtain the desired equality (4.30). Setting t = 0 in (4.27) and (4.33), we get(4.31) and (4.32) respectively. 2

Lemma 1.4.2 When V (x) ≡ 0 in the NLSE (4.1), we have

d2δV

dt2= 4E(ψ0) +

2β(dσ − 2)

σ + 1

∫

Rd|ψ2σ+2 dx. (4.35)

Proof: Sum (4.30) from j = 1 to d, we get

d2δV(t)

dt2=

d∑

j=1

d2δj(t)

dt2=

d∑

j=1

∫

Rd

(2|∂xjψ|2 +

2σβ

σ + 1|ψ|2σ+2

)dx

=

∫

Rd

[2|∇ψ|2 +

2dσβ

σ + 1|ψ|2σ+2

]dx

= 4E +2β(σd − 2)

σ + 1

∫

Rd|ψ|2σ+2dx. (4.36)

Here we use conservation of energy of the NLSE. 2

From this lemma, when V (x) ≡ 0 and at critical dimension, i.e. dσ − 2 = 0, (4.35)reduces to

d2δV

dt2= 4E, (4.37)

leading toδ

V(t) = 2Et2 + δ′

V(0)t+ δ

V(0). (4.38)

When the external potential V (x) is chosen as harmonic oscillator (3.13) and σ = 1 in(4.1), we have


Lemma 1.4.3 (i) In 1D without interaction, i.e. d = 1 and β = 0 in (4.1), we have

δx(t) =E(ψ0)

γ2x

+

(δ(0)x − E(ψ0)

γ2x

)cos(2γxt) +

δ(1)x

2γxsin(2γxt), t ≥ 0. (4.39)

(ii) In 2D with radial symmetry, i.e. d = 2 and γx = γy := γr in (4.1), for any initialdata ψ0(x, y) in (4.2), we have

δr(t) =E(ψ0)

γ2r

+

(δ(0)r − E(ψ0)

γ2r

)cos(2γrt) +

δ(1)r

2γrsin(2γrt), t ≥ 0, (4.40)

where

δr(t) = δx(t) + δy(t), δ(0)r := δr(0) = δx(0) + δy(0), δ(1)r := δ′r(0) = δ′x(0) + δ′y(0).

Furthermore, when d = 2 and γx = γy in (4.1) and the initial data ψ0(x) in (4.2) satisfying

ψ0(x, y) = f(r)eimθ with m ∈ Z and f(0) = 0 when m 6= 0, (4.41)

we have

δx(t) = δy(t) =1

2δr(t)

=E(ψ0)

2γ2x

+

(δ(0)x − E(ψ0)

2γ2x

)cos(2γxt) +

δ(1)x

2γxsin(2γxt), t ≥ 0. (4.42)

(iiii) For all other cases, we have

δj(t) =E(ψ0)

γ2xj

+

(δ(0)j − E(ψ0)

γ2xj

)cos(2γxj t) +

δ(1)j

2γxj

sin(2γxj t) + gj(t), t ≥ 0, (4.43)

where gj(t) is a solution of the following problem

d2gj(t)

dt2+ 4γ2

xjgj(t) = fj(t), gj(0) =

dgj(0)

dt= 0, (4.44)

with

fj(t) =

∫

Rd

[2|∂xjψ|2 − 2|∇ψ|2 − β|ψ|4 + (2γ2

xjx2j − 4V (x))|ψ|2

]dx

satisfying

|fα(t)| < 4Eβ(ψ0), t ≥ 0.

Proof: (i) From (4.30) with d = 1 and β1 = 0, we have

d2δx(t)

dt2= 4E(ψ0) − 4γ2

xδx(t), t > 0, (4.45)

δx(0) = δ(0)x , δ′x(0) = δ(1)x . (4.46)

Thus (4.39) is the unique solution of this ordinary differential equation (ODE).


(ii). From (4.30) with d = 2, we have

d2δx(t)

dt2= −2γ2

xδx(t) +

∫

Rd

(2|∂xψ|2 + β|ψ|4

)dx, (4.47)

d2δy(t)

dt2= −2γ2

yδy(t) +

∫

Rd

(2|∂yψ|2 + β|ψ|4

)dx. (4.48)

Sum (4.47) and (4.48), notice (4.4) and γx = γy, we have the ODE for δr(t):

d2δr(t)

dt2= 4E(ψ0) − 4γ2

xδr(t), t > 0, (4.49)

δr(0) = 2δ(0)x , δ′r(0) = 2δ(1)x . (4.50)

Thus (4.40) is the unique solution of the second order ODE (4.49) with the initial data(4.50). Furthermore, when the initial data ψ0(x) in (4.2) satisfies (4.41), due to the radialsymmetry, the solution ψ(x, t) of (4.1)-(4.2) satisfies

ψ(x, y, t) = g(r, t)eimθ with g(r, 0) = f(r). (4.51)

This implies

δx(t) =

∫

R2x2 |ψ(x, y, t)|2 dx =

∫ ∞

0

∫ 2π

0r2 cos2 θ |g(r, t)|2r dθdr

= π

∫ ∞

0r2|g(r, t)|2r dr =

∫ ∞

0

∫ 2π

0r2 sin2 θ |g(r, t)|2r dθdr

=

∫

R2y2|ψ(x, y, t)|2 dx = δy(t), t ≥ 0. (4.52)

Thus the equality (4.42) is a combination of (4.52) and (4.40).(iii). From (4.30), notice the energy conservation (4.4) of the GPE (4.1), we have

d2δj(t)

dt2= 4E(ψ0) − 4γ2

xjδj(t) + fj(t), t ≥ 0. (4.53)

Thus (4.43) is the unique solution of this ODE (4.53). 2

1.5 Plane and solitary wave solutions

For simplicity, we assume V (x) ≡ 0, d = 1 and σ = 1 in this section. In this case, theNLSE (4.1) collapses to

iψt = −1

2ψxx + β|ψ|2ψ. (5.1)

To find the plane wave solution of (5.1), we take the ansatz

ψ = Aei(kx−ωt), (5.2)

whereA, ω and k are amplitude, angular frequency and wavenumber respectively. Plugging(5.2) into (5.1), we get the dispersive relation

ω =1

2k2 + β|A|2 (5.3)

1.6. EXISTENCE/BLOWUP RESULTS 15

This implies that the dispersive relation depends on wavenumber and amplitude. Definethe group velocity

cg ≡dω

dk= k. (5.4)

So the NLSE has the plane wave solution (5.2) provided the dispersive relation is satisfied.In fact, (5.3) can be viewed as zeroth-order approximation of the NLSE (5.1), and (5.2)can be viewed as zeroth-order solution of the NLSE (5.1).

To find the solitary wave solution, we take the ansatz

ψ = φ(ξ)ei(kx−ωt), ξ = x− cgt, (5.5)

where φ is a real-valued function. Plugging (5.5) into (5.1), we get

1

2

d2φ

dξ2+ (ω − k2/2)φ − βφ3 + i(k − cg)

dφ

dξ= 0. (5.6)

This implies

−d2φ

dξ2+ γφ+ 2βφ3 = 0, γ = k2 − 2ω > 0; cg = k. (5.7)

When β < 0, we have a solution for (5.7)

φ(ξ) = ±√

γ

−β(2 − k2)dn

(√γ

2 − k2(ξ − ξ0), k

), (5.8)

where dn is the Jacobian elliptic function. Letting k → 1, we have

φ(ξ) = ±√

γ

−β sech√γ(ξ − ξ0). (5.9)

Thus we get a solitary wave solution for the NLSE (5.1) with β < 0:

ψ(x, t) =

√γ

−β sech√γ(x− t− x0)e

i[x−(1−γ)t/2], (5.10)

where γ > 0 is a constant.For β > 0, one can get a traveling wave in a similar manner.

1.6 Existence/blowup results

For simplicity, in this section, we assume V (x) ≡ 0 in (4.1).

1.6.1 Integral form

When β = 0 in (4.1), the free Schrodinger equation is solved as

ψ(x, t) = U(t)ψ0(x), x ∈ Rd, t ≥ 0, (6.1)

where free Schrodinger operator U(t) = eit4/2 given by

U(t)ψ0(x) =

(1

4πit

)d/2 ∫

Rdei

|x−x′ |2

4t ψ0(x′) dx′ (6.2)

defines a unitary transformation group in L2.


Theorem 1.6.1 (Decay estimates) For conjugate p and p′ (1p + 1

p′ = 1), with 2 ≤ p ≤ ∞,

and t 6= 0, the transformation U(t) maps continuously Lp′(Rd) into Lp(Rd) and

‖U(t)ψ0‖Lp ≤ (4π|t|)−d(12− 1

p)‖ψ0‖Lp′ . (6.3)

Proof (scratch): Use the conservation of L2-norm ‖ψ(t)‖L2 = ‖ψ0‖L2 , the estimate|ψ(x, t)| ≤ (4π|t|)−d/2‖ψ0‖L1 and the Riesz-Thorin interpolation theorem. 2

When β 6= 0 in (4.1), the problem is conveniently rewritten in the integral form

ψ(t) = U(t)ψ0 − iβ

∫ t

0U(t− t′)|ψ(t′)|2σψ(t′) dt′. (6.4)

1.6.2 Existence results

Based on a fixed point theorem to (6.4), the following existence results for NLSE is proved[113]:

Theorem 1.6.2 (Solution in H1) For 0 ≤ σ < 2d−2 (no condition on σ when d = 1

or 2) and an initial condition ψ0 ∈ H1(Rd), there exists, locally in time, a unique max-imal solution ψ in C((−T ∗, T ∗),H1(Rd)), where maximal means that if T ∗ < ∞, then‖ψ‖H1 → ∞ as t approaches T ∗. In addition, ψ satisfies the normalization and energy(or Hamiltonian) conservation laws

N(ψ) ≡∫

Rd|ψ|2 dx = N(ψ0), (6.5)

E(ψ) ≡∫

Rd

[1

2|∇ψ|2 +

β

σ + 1|ψ|2σ+2

]dx = E(ψ0), (6.6)

and depends continuously on the initial condition ψ in H1.If in addition, the initial condition ψ0 belongs to the space

∑= f, f ∈ H1(Rd), |xf(x)| ∈

L2(Rd) of the functions in H1 with finite variance, the above maximal solution belongsto C((−T ∗, T ∗),

∑). The variance δ

V(t) =

∫Rd |x|2|ψ|2 dx belongs to C2(−T ∗, T ∗) and

satisfies the identity

d2δV

dt2= 4E(ψ0) +

2β(dσ − 2)

σ + 1

∫

Rd|ψ|2σ+2 dx. (6.7)

Theorem 1.6.3 (Solution in L2) For 0 ≤ σ < 2d and an initial condition ψ0 ∈ L2(Rd),

there exist a unique solution ψ in C((−T ∗, T ∗), L2(Rd)) ∩ Lq((−T ∗, T ∗), L2σ+2(Rd)) with

q = 4(σ+1)dσ , satisfying the L2-norm conservation law (6.5).

Theorem 1.6.4 (Global existence in H1) Assume 0 ≤ σ < 2/(d − 2) if β < 0 (attractivenonlinearity), or 0 ≤ σ < 2/d if β > 0 (repulsive nonlinearity). For any ψ ∈ H1(Rd),there exists a unique solution ψ in C(R,H1(Rd)). It satisfies the conservation laws (6.5)and (6.6) and depends continuously on initial conditions in H1(Rd).

Theorem 1.6.5 (Global existence in L2) For 0 ≤ σ < 2/d and ψ0 ∈ L2(Rd), there existsa unique solution ψ in C(R, L2(Rd)) ∩ Lqloc(R, L

2σ+2(Rd)) with q = 4(σ + 1)/dσ thatsatisfies the L2-norm conservation (6.5) and depends continuously on initial conditions inL2.

1.7. WKB EXPANSION AND QUANTUM HYDRODYNAMICS 17

1.6.3 Finite time blowup results

Classical blowup results are based on the “variance identity”, also known as the “viraltheorem”, and “uncertainty principle”. Define the variance δ

V(t) =

∫Rd |x|2|ψ|2 dx, we

have the identityd2

dt2δV (t) = 4E +

2β(dσ − 2)

σ + 1

∫

Rd|ψ|2σ+2 dx. (6.8)

Theorem 1.6.6 Suppose that β < 0 and dσ ≥ 2. Consider an initial condition ψ0 ∈ H1

with δV(0) finite that satisfies one of the conditions below:

(i) E(ψ0) < 0,(ii) E(ψ0) = 0 and δ′

V(0) = 2 Re

∫Rd ψ∗

0(x · ∇ψ0)dx < 0,

(iii) E(ψ0) > 0 and∣∣∣δ′

V(0)∣∣∣ ≥ 2

√2E(ψ0)δV (0) = 2

√2E(ψ0)‖xψ0‖L2 .

Then, there exists a time t∗ <∞ such that

limt→t∗

‖∇ψ‖L2 = ∞ and limt→t∗

‖ψ‖L∞ = ∞. (6.9)

Proof: If β < 0 and dσ ≥ 2,d2

dt2δ

V(t) ≤ 4E, (6.10)

and by time integration,δ

V(t) ≤ 2Et2 + δ′

V(0)t+ δ

V(0). (6.11)

Under any of the hypotheses (i)-(iii) of the above theorem, there exists a time t0 such thatthe right-hand side of (6.11) vanishes, and thus also t1 ≤ t0 such that

limt→t1

δV(t) = 0. (6.12)

Furthermore, from the equality

∫

Rd|f |2 dx =

1

d

∫

Rd(∇ · x)|f |2 dx = −1

d

∫

Rdx · ∇(|f |2) dx, (6.13)

one gets the “uncertainty principle”

‖f‖2L2 ≤ 2

d‖∇f‖L2 ‖xf‖L2 . (6.14)

When this inequality is applied to a solution ψ, one gets from (6.14) and from the con-servation of ‖ψ‖2

L2 , that there exists a time t∗ ≤ t1 such that limt→t∗ ‖∇ψ‖L2 = ∞. Theconservation of E then ensures that limt→t∗ ‖ψ‖2σ+2

L2σ+2 = ∞, and since ‖ψ‖2L2 is conserved,

this implies that limt→t∗ ‖ψ‖L∞ = ∞. 2

1.7 WKB expansion and quantum hydrodynamics

In this section, we consider the NLSE in semiclassical regime

iεψεt = −ε2

24 ψε + V (x)ψε + f(|ψε|2)ψε, x ∈ Rd, t ≥ 0, (7.1)

ψε(x, 0) = ψε0(x), x ∈ Rd, (7.2)


where 0 < ε 1 is the (scaled) Planck constant, f(ρ) is a given real-valued function; andfind its semiclassical limit by using WKB expansion.

Suppose that the initial datum ψε0 in (7.2) is rapidly oscillating on the scale ε, givenin WKB form:

ψε0(x) = A0(x) exp

(i

εS0(x)

), x ∈ Rd, (7.3)

where the amplitude A0 and the phase S0 are smooth real-valued functions. Plugging theradial-representation of the wave-function

ψε(x, t) = Aε(x, t) exp

(i

εSε(x, t)

)=√ρε(x, t) exp

(i

εSε(x, t)

)(7.4)

into (7.1), one obtains the following quantum hydrodynamic (QHD) form of the NLSE forρε = |Aε|2, Jε = ρε∇Sε [58, 41, 79]

ρεt + div Jε = 0, (7.5)

Jεt + div

(Jε ⊗ Jε

ρε

)+ ∇P (ρε) + ρε∇V =

ε2

4div(ρε∇2 log ρε); (7.6)

with initial data

ρε(x, 0) = ρε0(x) = |A0(x)|2, Jε(x, 0) = ρε0(x)∇S0(x) = |A0(x)|2 ∇S0(x), (7.7)

(see Grenier [66], Jungel [79, 80], for mathematical analyses of this system). Here thehydrodynamic pressure P (ρ) is related to the nonlinear potential f(ρ) by

P (ρ) = ρf(ρ) −∫ ρ

0f(s) ds, (7.8)

i.e. f ′ is the enthalpy. Letting ε→ 0+, one obtains formally the following Euler system

ρt + div J = 0, (7.9)

Jt + div

(J⊗ J

ρ

)+ ∇P (ρ) + ρ∇V = 0. (7.10)

which can be viewed formally as the dispersive (semiclassical) limit of the NLSE (7.1). Inthe case f ′ > 0 we expect (7.9), (7.10) to be the ‘rigorous’ semiclassical limit of (7.1) aslong as caustics do not occur, i.e. in the pre-breaking regime. After caustics the dispersivebehavior of the NLSE takes over and (7.9), (7.10) is not correct any more.

1.8 Wigner transform and semiclassical limit

In this section, we consider the linear Schrodinger equation in semiclassical regime

iεψεt = −ε2

24 ψε + V (x)ψε, x ∈ Rd, t ≥ 0, (8.1)

ψε(x, 0) = ψε0(x), x ∈ Rd, (8.2)

and find its semiclassical limit by using Wigner transformation.

1.8. WIGNER TRANSFORM AND SEMICLASSICAL LIMIT 19

Let f, g ∈ L2(Rd). Then the Wigner-transform of (f, g) on the scale ε > 0 is definedas the phase-space function:

wε(f, g)(x, ξ) =1

(2π)d

∫

Rdf∗(x +

ε

2y

)g

(x− ε

2y

)eiy·ξ dy (8.3)

(cf. [58], [91] for a detailed analysis of the Wigner-transform). It is well-known that theestimate

‖wε(f, g)‖E∗ ≤ ‖f‖L2(Rd) ‖g‖L2(Rd) (8.4)

holds, where E is the Banach space

E := φ ∈ C0(Rdx × Rd

ξ) : (Fξ→vφ)(x,v) ∈ L1(Rdv;C0(R

dx)),

‖φ‖E :=

∫

Rdv

supx∈Rd

x

|(Fξ→vφ)(x,v)| dv,

(cf. [91]). E∗ denotes the dual space of E and (Fξ→vσ)(v) :=∫Rd

ξσ(ξ) e−iv·ξ dξ the

Fourier transform.

Now let ψε(t) be the solution of the linear Schrodinger equation (8.1), (8.2) and denote

wε(t) := wε(ψε(t), ψε(t)). (8.5)

Then wε satisfies the Wigner equation

wεt + ξ · ∇xwε + Θε[V ]wε = 0, (x, ξ) ∈ Rd

x × Rdξ , t ∈ R, (8.6)

wε(t = 0) = wε(ψε0, ψε0), (8.7)

where Θε[V ] is the pseudo-differential operator:

Θε[V ]wε(x, ξ, t) :=i

(2π)d

∫

Rdα

V (x + ε2α) − V (x − ε

2α)

εwε(x, α, t)eiα·ξdα, (8.8)

here wε stands for the Fourier-transform

Fξ→α wε(x, ·, t) :=

∫

Rdξ

wε(x, ξ, t)e−iα·ξ dξ.

The main advantage of the formulation (8.6), (8.7) is that the semiclassical limit ε → 0can easily be carried out. Taking ε to 0 gives the Vlasov-equation ( or Liouville equation):

w0t + ξ · ∇xw

0 −∇xV (x) · ∇ξw0 = 0, (x, ξ) ∈ Rd

x × Rdξ , t ∈ R, (8.9)

w0(t = 0) = w0 := limε→0

wε(ψε0, ψε0), (8.10)

(cf. [58], [91]), where

w0 := limε→0

wε.

Here, the limits hold in an appropriate weak sense (i.e. in E∗ − ω∗) and have to beunderstood for subsequences (εnk

) → 0 of sequence εn. We recall that w0, w0(t) are

positive bounded measures on the phase-space Rdx × Rd

ξ .


When the initial Wigner distribution has the high frequency form

w0 = |A0(x)|2δ(ξ −∇S0(x)), (8.11)

then it is easy to see that the solution of (8.9) is given that

w0(x, ξ, t) = |A(x, t)|2δ(ξ −∇S(x, t)), (8.12)

where A(x, t) is the solution of the transport equation

(|A|2)t + ∇ · (|A|2∇S) = 0, |A(x, 0)|2 = |A0(x)|2 (8.13)

and S(x, t) is the solution of the Eiconal equation

St +1

2|∇S|2 + V (x) = 0, S(x, 0) = S0(x). (8.14)

Define the moments

ρ(x, t) =

∫

Rdξ

w0(x, ξ, t) dξ, (8.15)

J(x, t) =

∫

Rdξ

ξw0(x, ξ, t) dξ. (8.16)

Then ρ and J satisfy the pressureless Euler equation:

ρt + div J = 0, (8.17)

Jt + div

(J⊗ J

ρ

)+ ρ∇V = 0; (8.18)

with initial data

ρ(x, 0) = ρ0(x) = |A0(x)|2, J(x, 0) = ρ0(x)∇S0(x) = |A0(x)|2 ∇S0(x). (8.19)

1.9 Ground, excited and central vortex states

For simplicity, in this section, we take σ = 1 and the potential V (x) as a harmonicoscillator (4.1), i.e. NLSE is considered in terms of BEC setup.

1.9.1 Stationary states

To find a stationary solution of (4.1), we write

ψ(x, t) = e−iµtφ(x), (9.1)

where µ is the chemical potential and φ is a function independent of time. Inserting (9.1)into (4.1) gives the following equation for φ(x)

µ φ(x) = −1

24 φ(x) + V (x) φ(x) + β|φ(x)|2φ(x), x ∈ Rd, (9.2)

1.9. GROUND, EXCITED AND CENTRAL VORTEX STATES 21

under the normalization condition

‖φ‖2 =

∫

Rd|φ(x)|2 dx = 1. (9.3)

This is a nonlinear eigenvalue problem under a constraint and any eigenvalue µ can becomputed from its corresponding eigenfunction φ by

µ = µ(φ) =

∫

Rd

[1

2|∇φ(x)|2 + V (x) |φ(x)|2 + β |φ(x)|4

]dx

= E(φ) +

∫

Rd

β

2|φ(x)|4 dx. (9.4)

In fact, the eigenfunctions of (9.2) under the constraint (9.3) are equivalent to the criticalpoints of the energy functional over the unit sphere S = φ | ‖φ‖ = 1, E(φ) < ∞.Furthermore, as noted in [6], they are equivalent to the steady state solutions of thefollowing continuous normalized gradient flow (CNGF):

∂tφ =1

24 φ− V (x)φ − β |φ|2φ+

µ(φ)

‖φ(·, t)‖2φ, x ∈ Rd, t ≥ 0, (9.5)

φ(x, 0) = φ0(x), x ∈ Rd with ‖φ0‖ = 1. (9.6)

1.9.2 Ground state

The BEC ground state wave function φg(x) is found by minimizing the energy functionalE(φ) over the unit sphere S = φ | ‖φ‖ = 1, E(φ) <∞:

(V) Find (µgβ , φgβ ∈ S) such that

Egβ = E(φgβ) = minφ∈S

E(φ), µgβ = µ(φgβ) = E(φgβ) +

∫

Rd

β

2|φgβ |2 dx. (9.7)

In the case of a defocusing condensate, i.e. β ≥ 0, the energy functional E(φ) is positive,coercive and weakly lower semicontinuous on S, thus the existence of a minimum followsfrom the standard theory. For understanding the uniqueness question note that E(αφgβ) =

E(φgβ) for all α ∈ C with |α| = 1. Thus an additional constraint has to be introduced toshow uniqueness. For non-rotating BECs, the minimization problem (9.7) has a uniquereal valued nonnegative ground state solution φgβ(x) > 0 for x ∈ Rd [87].

When β = 0, the ground state solution is given explicitly [5]

µg0 =1

2

γxγx + γyγx + γy + γz

φg0(x) =1

πd/4

γ1/4x e−γxx2/2, d = 1,

(γxγy)1/4e−(γxx2+γyy2)/2, d = 2,

(γxγyγz)1/4e−(γxx2+γyy2+γzz2)/2, d = 3.

(9.8)

In fact, this solution can be viewed as an approximation of the ground state for weaklyinteracting condensate, i.e. |βd| 1. For a condensate with strong repulsive interaction,i.e. β 1 and γα = O(1) (α = x, y, z), the ground state can be approximated by theThomas-Fermi approximation in this regime [5]:

φTFβ (x) =

√(µTFβ − V (x))/β, V (x) < µTF

β ,

0, otherwise,(9.9)

µTFβ =

1

2

(3βγx/2)2/3, d = 1,

(4βγxγy/π)1/2, d = 2,

(15βγxγyγz/4π)2/5, d = 3.

(9.10)


Due to φTFβ is not differentiable at V (x) = µTF

β , as noticed in [5, 8], E(φTFβ ) = ∞ and

µ(φTFβ ) = ∞. This shows that we can’t use (4.4) to define the energy of the Thomas-Fermi

approximation (9.9). How to define the energy of the Thomas-Fermi approximation is notclear in the literatures. Using (9.4), (9.10) and (9.9), here we present a way to define theenergy of the Thomas-Fermi approximation (9.9):

ETFβ = µTF

β −∫

Rd

β

2|φTFβ (x)|4 dx =

∫

Rd

[V (x)|φTF

β (x)|2 +β

2|φTFβ (x)|4

]dx

=d+ 2

d+ 4µTFβ , d = 1, 2, 3. (9.11)

From the numerical results in [6, 5], when γx = O(1), γy = O(1) and γz = O(1), we canget

Egβ − ETFβ = E(φgβ) −ETF

β → 0, as βd → ∞.

Any eigenfunction φ(x) of (9.2) under constraint (9.3) whose energy E(φ) > E(φgβ) isusually called as excited states in physical literatures.

1.9.3 Central vortex states

To find central vortex states in 2D with radial symmetry, i.e. d = 2 and γx = γy = 1 in(4.1), we write

ψ(x, t) = e−iµmtφm(x, y) = e−iµmtφm(r)eimθ, (9.12)

where (r, θ) is the polar coordinate, m 6= 0 is an integer and called as index or windingnumber, µm is the chemical potential, and φm(r) is a real function independent of time.Inserting (9.12) into (4.1) gives the following equation for φm(r)

µm φm(r) =

[− 1

2r

d

dr

(rd

dr

)+

1

2

(r2 +

m2

r2

)+ β2|φm|2

]φm, 0 < r <∞, (9.13)

φm(0) = 0, limr→∞

φm(r) = 0. (9.14)


2π

∫ ∞

0|φm(r)|2 r dr = 1. (9.15)

In order to find the central vortex state φmβ (x, y) = φmβ (r)eimθ with index m, we find areal nonnegative function φmβ (r) which minimizes the energy functional

Em(φ(r)) = E(φ(r)eimθ) = π

∫ ∞

0

[|φ′(r)|2 +

(r2 +

m2

r2

)|φ(r)|2 + β2|φ(r)|4

]rdr, (9.16)

over the set S0 = φ | 2π∫∞0 |φ(r)|2r dr = 1, φ(0) = 0, Em(φ) < ∞. The existence

and uniqueness of nonnegative minimizer for this minimization problem can be obtainedsimilarly as for the ground state [87]. Note that the set Sm = φ(r)eimθ | φ ∈ S0 ⊂ S isa subset of the unit sphere, so φmβ (r)eimθ is a minimizer of the energy functional Eβ over

the set Sm ⊂ S. When β2 = 0 in (4.1), φm0 (r) = 1√π|m|!

r|m|e−r2/2 [6].

1.9. GROUND, EXCITED AND CENTRAL VORTEX STATES 23

Similarly, in order to find central vortex line states in 3D with cylindrical symmetry,i.e. d = 3 and γx = γy = 1 in (4.1), we write

ψ(x, t) = e−iµmtφm(x, y, z) = e−iµmtφm(r, z)eimθ , (9.17)

where m 6= 0 is an integer and called as index, µm is the chemical potential, and φm(r, z)is a real function independent of time. Inserting (9.17) into (4.1) with d = 3 gives thefollowing equation for φm(r, z)

µm φm =

[− 1

2r

∂

∂r

(r∂

∂r

)− ∂2

2 ∂z2+

1

2

(r2 +

m2

r2+ γ2

zz2

)+ β3|φm|2

]φm, (9.18)

φm(0, z) = 0, limr→∞

φm(r, z) = 0, −∞ < z <∞, (9.19)

lim|z|→∞

φm(r, z) = 0, 0 ≤ r <∞, (9.20)


2π

∫ ∞

0

∫ ∞

−∞|φm(r, z)|2 r drdz = 1. (9.21)

In order to find the central vortex line state φmβ (x, y, z) = φmβ (r, z)eimθ with index m, wefind a real nonnegative function φmβ (r, z) which minimizes the energy functional

Em(φ(r, z)) = E(φ(r, z)eimθ) (9.22)

= π

∫ ∞

0

∫ ∞

−∞

[|∂rφ|2 + |∂zφ|2 +

(r2 + γ2

zz2 +

m2

r2

)|φ|2 + β3|φ|4

]r drdz,

over the set S0 = φ | 2π∫∞0

∫∞−∞ |φ(r, z)|2r drdz = 1, φ(0, z) = 0, −∞ < z <

∞, Emβ (φ) < ∞. The existence and uniqueness of nonnegative minimizer for this min-imization problem can be obtained similarly as for the ground state [87]. Note that theset Sm = φ(r, z)eimθ | φ ∈ S0 ⊂ S is a subset of the unit sphere, so φmβ (r, z)eimθ

is a minimizer of the energy functional Eβ over the set Sm. When β3 = 0 in (3.11),

φm0 (r, z) = γ1/4z

π3/4√

|m|!r|m|e−(r2+γzz2)/2 [6].

1.9.4 Variation of stationary states over the unit sphere

For the stationary states of (9.2), we have the following lemma:

Lemma 1.9.1 Suppose β = 0 and V (x) ≥ 0 for x ∈ Rd, we have(i) The ground state φg is a global minimizer of E(φ) over S.(ii) Any excited state φj is a saddle point of E(φ) over S.

Proof: Let φe be an eigenfunction of the eigenvalue problem (9.2) and (9.3). The cor-responding eigenvalue is µe. For any function φ such that E(φ) < ∞ and ‖φe + φ‖ = 1,notice (9.3), we have that

‖φ‖2 = ‖φ+ φe − φe‖2 = ‖φ+ φe‖2 − ‖φe‖2 −∫

Rd(φ∗φe + φφ∗e) dx

= −∫

Rd(φ∗φe + φφ∗e) dx. (9.23)


From (4.4) with ψ = φe + φ and β = 0, notice (9.3) and (9.23), integration by parts, weget

E(φe + φ) =

∫

Rd

[1

2|∇φe + ∇φ|2 + V (x)|φe + φ|2

]dx

=

∫

Rd

[1

2|∇φe|2 + V (x)|φe|2

]+

∫

Rd

[1

2|∇φ|2 + V (x)|φ|2

]dx

+

∫

Rd

[(−1

24 φe + V (x)φe

)∗

φ+

(−1

24 φe + V (x)φe

)φ∗]dx

= E(φe) + E(φ) +

∫

Rd(µeφ

∗e + µeφeφ

∗) dx

= E(φe) + E(φ) − µe‖φ‖2 = E(φe) + [E(φ/‖φ‖) − µe] ‖φ‖2. (9.24)

(i) Taking φe = φg and µe = µg in (9.24) and noticing E(φ/‖φ‖) ≥ E(φg) = µg forany φ 6= 0, we get immediately that φg is a global minimizer of E(φ) over S.

(ii). Taking φe = φj and µe = µj in (9.24), since E(φg) < E(φj) and it is easy tofind an eigenfunction φ of (9.2) such that E(φ) > E(φj), we get immediately that φj is asaddle point of the functional E(φ) over S. 2

1.9.5 Conservation of angular momentum expectation

Another important quantity for studying dynamics of BEC in 2&3d, especially for mea-suring the appearance of vortex, is the angular momentum expectation value defined as

〈Lz〉(t) :=

∫

Rdψ∗(x, t)Lzψ(x, t) dx, t ≥ 0, d = 2, 3, (9.25)

where Lz = i (y∂x − x∂y) is the z-component angular momentum.

Lemma 1.9.2 Suppose ψ(x, t) is the solution of the problem (4.1), (4.2) with d = 2 or 3,then we have

d〈Lz〉(t)dt

=(γ2x − γ2

y

)δxy(t), δxy(t) =

∫

Rdxy|ψ(x, t)|2 dx, t ≥ 0. (9.26)

This implies that, at least in the following two cases, the angular momentum expectationis conserved:

i) For any given initial data ψ0(x) in (4.2), if the trap is radial symmetric in 2d, andresp., cylindrical symmetric in 3d, i.e. γx = γy;

ii) For any given γx > 0 and γy > 0 in (3.13), if the initial data ψ0(x) in (4.2) iseither odd or even in the first variable x or second variable y.

Proof: Differentiate (9.25) with respect to t, notice (4.1), integrate by parts, we have

d〈Lz〉(t)dt

=

∫

Rd[(iψ∗

t ) (y∂x − x∂y)ψ + ψ∗ (y∂x − x∂y)(iψt)] dx

=

∫

Rd

[(1

2∇2ψ∗ − V (x)ψ∗ − β|ψ|2ψ∗

)(y∂x − x∂y)ψ

+ψ∗ (y∂x − x∂y)

(−1

2∇2ψ + V (x)ψ + β|ψ|2ψ

)]dx

1.10. NUMERICAL METHODS FOR COMPUTING GROUND STATES 25

=

∫

Rd

1

2

[∇2ψ∗ (y∂x − x∂y)ψ − ψ∗ (y∂x − x∂y)∇2ψ

]dx +

∫

Rd

[ψ∗ (y∂x − x∂y)

(V (x)ψ + β|ψ|2ψ

)

−(V (x)ψ∗ + β|ψ|2ψ∗

)(y∂x − x∂y)ψ

]dx

=

∫

Rd|ψ|2(y∂x − x∂y)

(V (x) + β|ψ|2

)dx

=

∫

Rd|ψ|2(y∂x − x∂y)Vd(x) dx =

∫

Rd|ψ|2(γ2

x − γ2y)xy dx

= (γ2x − γ2

y)

∫

Rdxy|ψ|2 dx, t ≥ 0. (9.27)

For case i), since γx = γy, we get the conservation of 〈Lz〉 immediately from the firstorder ODE:

d〈Lz〉(t)dt

= 0, t ≥ 0. (9.28)

For case ii), we know the solution ψ(x, t) is either odd or even in the first variable x orsecond variable y due to the assumption of the initial data and symmetry of V (x). Thus|ψ(x, t)| is even in either x or y, which immediately implies that 〈Lz〉 satisfies the firstorder ODE (9.28). 2

1.10 Numerical methods for computing ground states

In this section, we present the continuous normalized gradient flow (CNGF), prove itsenergy diminishing and propose its semi-discretization for computing ground states inBEC. For simplicity, we take σ = 1 in (4.1).

1.10.1 Gradient flow with discrete normalization (GFDN)

Various algorithms for computing the minimizer of the energy functional E(φ) under theconstraint (9.3) have been studied in the literature. For instance, second order in timediscretization scheme that preserves the normalization and energy diminishing propertieswere presented in [2, 6]. Perhaps one of the more popular technique for dealing withthe normalization constraint (9.3) is through the following construction: choose a timesequence 0 = t0 < t1 < t2 < · · · < tn < · · · with 4tn = tn+1−tn > 0 and k = maxn≥0 4tn.To adapt an algorithm for the solution of the usual gradient flow to the minimizationproblem under a constraint, it is natural to consider the following splitting (or projection)scheme which was widely used in physical literatures [6] for computing the ground statesolution of BEC:

φt = −1

2

δE(φ)

δφ=

1

24 φ− V (x)φ− β |φ|2φ, x ∈ Ω, tn < t < tn+1, n ≥ 0, (10.1)

φ(x, tn+1)4= φ(x, t+n+1) =

φ(x, t−n+1)

‖φ(·, t−n+1)‖, x ∈ Ω, n ≥ 0, (10.2)

φ(x, t) = 0, x ∈ Γ = ∂Ω, φ(x, 0) = φ0(x), x ∈ Ω; (10.3)

where φ(x, t±n ) = limt→t±nφ(x, t), ‖φ0‖ = 1 and Ω ⊂ Rd. In fact, the gradient flow (10.1)

can be viewed as applying the steepest decent method to the energy functional E(φ)


without constraint and (10.2) then projects the solution back to the unit sphere in orderto satisfying the constraint (9.3). From the numerical point of view, the gradient flow(10.1) can be solved via traditional techniques and the normalization of the gradient flowis simply achieved by a projection at the end of each time step.

1.10.2 Energy diminishing of GFDN

Let

φ(·, t) =φ(·, t)‖φ(·, t)‖ , tn ≤ t ≤ tn+1, n ≥ 0. (10.4)

For the gradient flow (10.1), it is easy to establish the following basic facts:

Lemma 1.10.1 Suppose V (x) ≥ 0 for all x ∈ Ω, β ≥ 0 and ‖φ0‖ = 1, then(i). ‖φ(·, t)‖ ≤ ‖φ(·, tn)‖ = 1 for tn ≤ t ≤ tn+1, n ≥ 0.(ii). For any β ≥ 0,

E(φ(·, t)) ≤ E(φ(·, t′)), tn ≤ t′ < t ≤ tn+1, n ≥ 0. (10.5)

(iii). For β = 0,

E(φ(·, t)) ≤ E(φ(·, tn)), tn ≤ t ≤ tn+1, n ≥ 0. (10.6)

Proof: (i) and (ii) follows the standard techniques used for gradient flow. As for (iii),from (4.4) with ψ = φ and β = 0, (10.1), (10.3) and (10.4), integration by parts andSchwartz inequality, we obtain

d

dtE(φ) =

d

dt

∫

Ω

[|∇φ|22‖φ‖2

+V (x)φ2

‖φ‖2

]dx

= 2

∫

Ω

[∇φ · ∇φt2‖φ‖2

+V (x)φ φt‖φ‖2

]dx−

(d

dt‖φ‖2

) ∫

Ω

[|∇φ|22‖φ‖4

+V (x)φ2

‖φ‖4

]dx

= 2

∫

Ω

[−1

2 4 φ+ V (x)φ]φt

‖φ‖2dx−

(d

dt‖φ‖2

)∫

Ω

12 |∇φ|2 + V (x)φ2

‖φ‖4dx

= −2‖φt‖2

‖φ‖2+

1

2‖φ‖4

(d

dt‖φ‖2

)2

=2

‖φ‖4

[(∫

Ωφ φt dx

)2

− ‖φ‖2‖φt‖2

]

≤ 0 , tn ≤ t ≤ tn+1. (10.7)

This implies (10.6). 2

Remark 1.10.1 The property (10.5) is often referred as the energy diminishing propertyof the gradient flow. It is interesting to note that (10.6) implies that the energy diminishingproperty is preserved even with the normalization of the solution of the gradient flow forβ = 0, that is, for linear evolution equations.

Remark 1.10.2 When β > 0, the solution of (10.1)-(10.3) may not preserve the normal-ized energy diminishing property

E(φ(·, t)) ≤ E(φ(·, t′)), 0 ≤ t′ < t ≤ t1

for any t1 > 0 [6].


From Lemma 1.10.1, we get immediately

Theorem 1.10.1 Suppose V (x) ≥ 0 for all x ∈ Ω and ‖φ0‖ = 1. For β = 0, GFDN(10.1)-(10.3) is energy diminishing for any time step k and initial data φ0, i.e.

E(φ(·, tn+1)) ≤ E(φ(·, tn)) ≤ · · · ≤ E(φ(·, 0)) = E(φ0), n = 0, 1, 2, · · · . (10.8)

1.10.3 Continuous normalized gradient flow (CNGF)

In fact, the normalized step (10.2) is equivalent to solve the following ODE exactly

φt(x, t) = µφ(t, k)φ(x, t), x ∈ Ω, tn < t < tn+1, n ≥ 0, (10.9)

φ(x, t+n ) = φ(x, t−n+1), x ∈ Ω; (10.10)

where

µφ(t, k) ≡ µφ(tn+1,4tn) = − 1

2 4 tnln ‖φ(·, t−n+1)‖2, tn ≤ t ≤ tn+1. (10.11)

Thus the GFDN (10.1)-(10.3) can be viewed as a first-order splitting method for thegradient flow with discontinuous coefficients:

φt =1

24 φ− V (x)φ− β |φ|2φ+ µφ(t, k)φ, x ∈ Ω, t ≥ 0, (10.12)

φ(x, t) = 0, x ∈ Γ, φ(x, 0) = φ0(x), x ∈ Ω. (10.13)

Let k → 0, we see that

limk→0+

µφ(t, k) = µφ(t) =1

‖φ(·, t)‖2

∫

Ω

[1

2|∇φ(x, t)|2 + V (x)φ2(x, t) + βφ4(x, t)

]dx.

(10.14)This suggests us to consider the following continuous normalized gradient flow:

φt =1

24 φ− V (x)φ− β |φ|2φ+ µφ(t)φ, x ∈ Ω, t ≥ 0, (10.15)

φ(x, t) = 0, x ∈ Γ, φ(x, 0) = φ0(x), x ∈ Ω. (10.16)

In fact, the right hand side of (10.15) is the same as (9.2) if we view µφ(t) as a Lagrangemultiplier for the constraint (9.3). Furthermore for the above CNGF, as observed in [6],the solution of (10.15) also satisfies the following theorem:

Theorem 1.10.2 Suppose V (x) ≥ 0 for all x ∈ Ω, β ≥ 0 and ‖φ0‖ = 1. Then the CNGF(10.15)-(10.16) is normalization conservation and energy diminishing, i.e.

‖φ(·, t)‖2 =

∫

Ωφ2(x, t) dx = ‖φ0‖2 = 1, t ≥ 0, (10.17)

d

dtE(φ) = −2 ‖φt(·, t)‖2 ≤ 0 , t ≥ 0, (10.18)

which in turn implies

E(φ(·, t1)) ≥ E(φ(·, t2)), 0 ≤ t1 ≤ t2 <∞.


Remark 1.10.3 We see from the above theorem that the energy diminishing property ispreserved in the continuous dynamic system (10.15).

Using argument similar to that in [88, 106], we may also get as t → ∞, φ approachesto a steady state solution which is a critical point of the energy. In non-rotating BEC,it has a unique real valued nonnegative ground state solution φg(x) ≥ 0 for all x ∈ Ω[87]. We choose the initial data φ0(x) ≥ 0 for x ∈ Ω, e.g. the ground state solution oflinear Schrodinger equation with a harmonic oscillator potential [5, 8]. Under this kind ofinitial data, the ground state solution φg and its corresponding chemical potential µg canbe obtained from the steady state solution of the CNGF (10.15)-(10.16), i.e.

φg(x) = limt→∞

φ(x, t), x ∈ Ω, µg = µβ(φg) = E(φg) +β

2

∫

Ω|φg(x)|4 dx. (10.19)

1.10.4 Semi-implicit time discretization

To further discretize the equation (10.1), we here consider the following semi-implicit timediscretization scheme:

φn+1 − φn

k=

1

24 φn+1 − V (x)φn+1 − β |φn|2φn+1 , x ∈ Ω, (10.20)

φn+1(x) = 0, x ∈ Γ, φn+1(x) = φn+1(x)/‖φn+1‖ , x ∈ Ω . (10.21)

Notice that since the equation (10.20) becomes linear, the solution at the new timestep becomes relatively simple. In other words, in each discrete time interval, we mayview (10.20) as a discretization of a linear gradient flow with a modified potential Vn(x) =V (x) + β|φn(x)|2.

We now first present the following lemma:

Lemma 1.10.2 Suppose β ≥ 0 and V (x) ≥ 0 for all x ∈ Ω and ‖φn‖ = 1. Then,

∫

Ω|φn+1|2 dx ≤

∫

Ωφn φn+1 dx,

∫

Ω|φn+1|4 dx ≤

∫

Ω|φn|2 |φn+1|2 dx. (10.22)

Proof: Multiplying both sides of (10.20) by φn+1, integrating over Ω, and applyingintegration by parts, we obtain

∫

Ω

(|φn+1|2 − φnφn+1

)dx = −k

∫

Ω

[1

2|∇φn+1|2 + Vn(x)|φn+1|2

]dx ≤ 0 ,

which leads to the first inequality in (10.22). Similarly,

∫

Ω|φn+1|2|φn|2dx =

∫

Ω|φn+1|2

∣∣∣∣φn+1 − k

24 φn+1 + kVn(x)φn+1

∣∣∣∣2

dx

=

∫

Ω|φn+1|2

[|φn+1|2 − 2

k

2φn+1 4 φn+1 + 2kVn(x)|φn+1|2

]dx

+

∫

Ω|φn+1|2

∣∣∣∣k

24 φn+1 − kVn(x)φn+1

∣∣∣∣2

dx


=

∫

Ω|φn+1|2

[|φn+1|2 + 3k|∇φn+1|2 + 2kVn(x)|φn+1|2

]dx

+

∫

Ω|φn+1|2

∣∣∣∣k

24 φn+1 − kVn(x)φn+1

∣∣∣∣2

dx

≥∫

Ω|φn+1|4dx . (10.23)

This implies the second inequality in (10.22). 2

Given a linear self-adjoint operator A in a Hilbert space H with inner product (·, ·),and assume that A is positive definite in the sense that for some positive constant c,(u,Au) ≥ c(u, u) for any u ∈ H. We now present a simple lemma:

Lemma 1.10.3 For any k > 0, and (I + kA)u = v, we have

(u,Au)

(u, u)≤ (v,Av)

(v, v). (10.24)

Proof: Since A is self-adjoint and positive definite, by Holder inequality, we have for anyp, q ≥ 1 with p+ q = pq, that

(u,Au) ≤ (u, u)1/p (u,Aqu)1/q ,

which leads to

(u,Au) ≤ (u, u)1/2(u,A2u

)1/2, (u,Au)

(u,A2u

)≤ (u, u)

(u,A3u

).

Direct calculation then gives

(u,Au) ((I + kA)u, (I + kA)u)

= (u,Au) (u, u) + 2k (u,Au)2 + k2 (u,Au)(u,A2u

)

≤ (u,Au) (u, u) + 2k (u, u)(u,A2u

)+ k2 (u, u)

(u,A3u

)

= (u, u) ((I + kA)u,A(I + kA)u) . (10.25)

2

Let us define a modified energy Eφn as

Eφn(u) =

∫

Ω

[1

2|∇u|2 + Vn(x)|u|2

]dx =

∫

Ω

[1

2|∇u|2 + V (x)|u|2 + β|φn|2|u|2

]dx ,

we then get from the above lemma that

Lemma 1.10.4 Suppose V (x) ≥ 0 for all x ∈ Ω, β ≥ 0 and ‖φn‖ = 1. Then,

Eφn(φn+1) ≤ Eφn(φn+1)

‖φn+1‖= Eφn

(φn+1

‖φn+1‖

)= Eφn(φn+1) ≤ Eφn(φn) . (10.26)

Using the inequality (10.22), we in turn get:


Lemma 1.10.5 Suppose V (x) ≥ 0 for all x ∈ Ω and β ≥ 0, then,

E(φn+1) ≤ E(φn),

where

E(u) =

∫

Ω

[1

2|∇u|2 + V (x)|u|2 + β|u|4

]dx .

Remark 1.10.4 As we noted earlier, for β = 0, the energy diminishing property is pre-served in the GFDN (10.1)-(10.3) and semi-implicit time discretization (10.20)-(10.21).For β > 0, the energy diminishing property in general does not hold uniformly for all φ0

and all step size k > 0, a justification on the energy diminishing is presently only possiblefor a modified energy within two adjacent steps.

1.10.5 Discretized normalized gradient flow (DNGF)

Consider a discretization for the GFDN (10.20)-(10.21) (or a fully discretization of (10.15)-(10.16))

Un+1 − Un

k= −AUn+1, Un+1 =

Un+1

‖Un+1‖, n = 0, 1, 2, · · · ; (10.27)

where Un = (un1 , un2 , · · · , unM−1)

T , k > 0 is time step and A is an (M − 1) × (M − 1)symmetric positive definite matrix. We adopt the inner product, norm and energy ofvectors U = (u1, u2, · · · , uM−1)

T and V = (v1, v2, · · · , vM−1)T as

(U, V ) = UTV =M−1∑

j=1

uj vj, ‖U‖2 = UTU = (U,U), E(U) = UTAU = (U,AU),

(10.28)respectively. Using the finite dimensional version of the lemmas given in the previoussubsection, we have

Theorem 1.10.3 Suppose ‖U0‖ = 1 and A is symmetric positive definite. Then theDNGF (10.27) is energy diminishing, i.e.

E(Un+1

)≤ E (Un) ≤ · · · ≤ E

(U0), n = 0, 1, 2, · · · . (10.29)

Furthermore if I+kA is an M -matrix, then (I+kA)−1 is a nonnegative matrix (i.e. withnonnegative entries). Thus the flow is monotone, i.e. if U0 is a non-negative vector, thenUn is also a non-negative vector for all n ≥ 0.

Remark 1.10.5 If a discretization for the GFDN (10.20)-(10.21) reads

Un+1 − Un

k= −BUn, Un+1 =

Un+1

‖Un+1‖, n = 0, 1, 2, · · · . (10.30)

For symmetric, positive definite B with ρ(kB) < 1 (ρ(B) being the spectral radius of B),(10.29) is satisfied by choosing

A =1

k

((I − kB)−1 − I

)= (I − kB)−1B.



Un+1 = BUn, Un+1 =Un+1

‖Un+1‖, n = 0, 1, 2, · · · . (10.31)

For symmetric, positive definite B with ρ(B) < 1, (10.29) is satisfied by choosing

A =1

k

(B−1 − I

).


Un+1 − Un

k= −BUn+1 − CUn, Un+1 =

Un+1

‖Un+1‖, n = 0, 1, 2, · · · . (10.32)

Suppose B and C are symmetric, positive definite and ρ(kC) < 1. Then (10.29) is satisfiedby choosing

A = (I − kC)−1 (B +C).

1.10.6 Numerical methods

In this section, we will present two numerical methods to discretize the GFDN (10.1)-(10.3) (or a full discretization of the CNGF (10.15)-(10.16)). For simplicity of notationwe introduce the methods for the case of one spatial dimension (d = 1) with homogeneousperiodic boundary conditions. Generalizations to higher dimension are straightforward fortensor product grids and the results remain valid without modifications. For d = 1, wehave

φt =1

2φxx − V (x)φ− β |φ|2φ, x ∈ Ω = (a, b), tn < t < tn+1, n ≥ 0, (10.33)

φ(x, tn+1)4= φ(x, t+n+1) =

φ(x, t−n+1)

‖φ(·, t−n+1)‖, a ≤ x ≤ b, n ≥ 0, (10.34)

φ(x, 0) = φ0(x), a ≤ x ≤ b, φ(a, t) = φ(b, t) = 0, t ≥ 0; (10.35)

with

‖φ0‖2 =

∫ b

aφ2

0(x) dx = 1.

We choose the spatial mesh size h = 4x > 0 with h = (b − a)/M and M an evenpositive integer, the time step is given by k = 4t > 0 and define grid points and timesteps by

xj := a+ j h, tn := n k, j = 0, 1, · · · ,M, n = 0, 1, 2, · · ·

Let φnj be the numerical approximation of φ(xj , tn) and φn the solution vector at timet = tn = nk with components φnj .

Backward Euler finite difference (BEFD) We use backward Euler for time dis-cretization and second-order centered finite difference for spatial derivatives. The detail


scheme is:

φ∗j − φnjk

=1

2h2

[φ∗j+1 − 2φ∗j + φ∗j−1

]− V (xj)φ

∗j − β

(φnj

)2φ∗j , j = 1, · · · ,M − 1,

φ∗0 = φ∗M = 0, φ0j = φ0(xj), j = 0, 1, · · · ,M,

φn+1j =

φ∗j‖φ∗‖ , j = 0, · · · ,M, n = 0, 1, · · · ; (10.36)

where the norm is defined as ‖φ∗‖2 = h∑M−1j=1

(φ∗j

)2.

Time-splitting sine-spectral method (TSSP) From time t = tn to time t = tn+1,the equation (10.33) is solved in two steps. First, one solves

φt =1

2φxx, (10.37)

for one time step of length k, then followed by solving

φt(x, t) = −V (x)φ(x, t) − β|φ|2φ(x, t), tn ≤ t ≤ tn+1, (10.38)

again for the same time step. Equation (10.37) is discretized in space by the sine-spectralmethod and integrated in time exactly. For t ∈ [tn, tn+1], multiplying the ODE (10.38) byφ(x, t), one obtains with ρ(x, t) = φ2(x, t)

ρt(x, t) = −2V (x)ρ(x, t) − 2βρ2(x, t), tn ≤ t ≤ tn+1. (10.39)

The solution of the ODE (10.39) can be expressed as

ρ(x, t) =

V (x)ρ(x, tn)

(V (x) + βρ(x, tn)) e2V (x)(t−tn) − βρ(x, tn)V (x) 6= 0,

ρ(x, tn)

1 + 2βρ(x, tn)(t− tn), V (x) = 0.

(10.40)

Combining the splitting step via the standard second-order Strang splitting for solvingthe GFDN (10.33)-(10.35), in detail, the steps for obtaining φn+1

j from φnj are given by

φ∗j =

√√√√ V (xj)e−kV (xj)

V (xj) + β(1 − e−kV (xj))|φnj |2φnj V (xj) 6= 0,

1√1 + βk|φnj |2

φnj , V (xj) = 0,

φ∗∗j =M−1∑

l=1

e−kµ2l /2 φ∗l sin(µl(xj − a)), j = 1, 2, · · · ,M − 1,

φ∗∗∗j =

√√√√ V (xj)e−kV (xj)

V (xj) + β(1 − e−kV (xj))|φ∗∗j |2 φ∗∗j V (xj) 6= 0,

1√1 + βk|φ∗∗j |2

φ∗∗j , V (xj) = 0,

φn+1j =

φ∗∗∗j

‖φ∗∗∗‖ , j = 0, · · · ,M, n = 0, 1, · · · ; (10.41)


where Ul are the sine-transform coefficients of a real vector U = (u0, u1, · · · , uM )T withu0 = uM = 0 which are defined as

µl =πl

b− a, Ul =

2

M

M−1∑

j=1

uj sin(µl(xj − a)), l = 1, 2, · · · ,M − 1 (10.42)

andφ0j = φ(xj , 0) = φ0(xj), j = 0, 1, 2, · · · ,M.

Note that the only time discretization error of TSSP is the splitting error, which is secondorder in k.

For comparison purposes we review a few other numerical methods which are currentlyused for solving the GFDN (10.33)-(10.35). One is the Crank-Nicolson finite difference(CNFD) scheme:

φ∗j − φnjk

=1

4h2

[φ∗j+1 − 2φ∗j + φ∗j−1 + φnj+1 − 2φnj + φnj−1

]

−V (xj)

2

[φ∗j + φnj

]−β∣∣∣φnj∣∣∣2

2

[φ∗j + φnj

], j = 1, · · · ,M − 1,

φ∗0 = φ∗M = 0, φ0j = φ0(xj), j = 0, 1, · · · ,M,

φn+1j =

φ∗j‖φ∗‖ , j = 0, · · · ,M, n = 0, 1, · · · . (10.43)

Another one is the forward Euler finite difference (FEFD) method:

φ∗j − φnjk

=1

2h2

[φnj+1 − 2φnj + φnj−1

]− V (xj)φ

nj − β

∣∣∣φnj∣∣∣2φnj , j = 1, · · · ,M − 1,

φ∗0 = φ∗M = 0, φ0j = φ0(xj), j = 0, 1, · · · ,M,

φn+1j =

φ∗j‖φ∗‖ , j = 0, · · · ,M, n = 0, 1, · · · ; (10.44)

1.10.7 Energy diminishing of DNGF

First we analyze the energy diminishing of the different numerical methods for linear case,i.e. β = 0 in (10.33). Introducing

Φn =(φn1 , φ

n2 , · · · , φnM−1

)T,

D = (djl)(M−1)×(M−1) , with djl =1

2h2

2 j = l,−1 |j − l| = −1,0 otherwise,

j, l = 1, · · · ,M − 1,

E = diag (V (x1), V (x2), · · · , V (xM−1)) ,

F (Φ) = diag(φ2

1, φ22, · · · , φ2

M−1

), with Φ = (φ1, φ2, · · · , φM−1)

T ,

G = (gjl)(M−1)×(M−1) , with gjl =2

M

M−1∑

m=1

sinπmj

Msin

πml

Me−kµ

2m/2,

H = diag(e−kV (x1)/2, e−kV (x2)/2, · · · , e−kV (xM−1)/2

).


Then the BEFD discretization (10.36) (called as BEFD normalized flow) with β = 0 canbe expressed as

Φ∗ − Φn

k= −(D + E)Φ∗, Φn+1 =

Φ∗

‖Φ∗‖ , n = 0, 1, · · · . (10.45)

The TSSP discretization (10.41) (called as TSSP normalized flow) with β = 0 can beexpressed as

Φ∗∗∗ = HΦ∗∗ = HGΦ∗ = HGHΦn, Φn+1 =Φ∗

‖Φ∗‖ , n = 0, 1, · · · . (10.46)

The CNFD discretization (10.43) (called as CNFD normalized flow) with β = 0 can beexpressed as

Φ∗ − Φn

k= −1

2(D + E)Φ∗ − 1

2(D + E)Φn, Φn+1 =

Φ∗

‖Φ∗‖ , n = 0, 1, · · · . (10.47)

The FEFD discretization (10.44) (called as FEFD normalized flow) with β = 0 can beexpressed as

Φ∗ − Φn

k= −(D + E)Φn, Φn+1 =

Φ∗

‖Φ∗‖ , n = 0, 1, · · · . (10.48)

It is easy to see that D and G are symmetric positive definite matrices. FurthermoreD is also an M -matrix and ρ(D) =

(1 + cos π

M

)/h2 < 2/h2 and ρ(G) = e−kµ

21/2 < 1.

Applying the theorem 1.10.3 and remarks 1.10.5, 1.10.6 and 1.10.7, we have

Theorem 1.10.4 Suppose V ≥ 0 in Ω and β = 0. We have that(i). The BEFD normalized flow (10.36) is energy diminishing and monotone for any

k > 0.(ii). The TSSP normalized flow (10.41) is energy diminishing for any k > 0.(iii). The CNFD normalized flow (10.43) is energy diminishing and monotone provided

that

k ≤ 2

2/h2 + maxj V (xj)=

2h2

2 + h2 maxj V (xj). (10.49)

(iv). The FEFD normalized flow (10.44) is energy diminishing and monotone providedthat

k ≤ 1

2/h2 + maxj V (xj)=

h2

2 + h2 maxj V (xj). (10.50)

For nonlinear case, i.e. β > 0, we only analyze the energy between two steps of BEFDflow (10.36). In this case, consider

Φn+1 − Φn

k= − (D + E + βF (Φn)) Φn+1, Φn+1 =

Φn+1

‖Φn+1‖. (10.51)

Lemma 1.10.6 Suppose V ≥ 0, β > 0 and ‖Φn‖ = 1. Then for the flow (10.51), we have

E(Φn+1

)≤ E (Φn) , EΦn

(Φn+1

)≤ EΦn (Φn) (10.52)


where

E (Φ) = (Φ, (D + E + βF (Φ))Φ) = ΦT (D + E)Φ + βM−1∑

j=1

φ4j , (10.53)

EΦn (Φ) = (Φ, (D + E + βF (Φn))Φ) = ΦT (D + E)Φ + βM−1∑

j=1

φ2j

(φnj

)2. (10.54)

Proof: Combining (10.51), (10.27) and Theorem 1.10.3, we have

(Φn+1, (D + E + βF (Φn))Φn+1

)≤

(Φn+1, (D + E + βF (Φn))Φn+1

)

(Φn+1, Φn+1

)

≤ (Φn, (D + E + βF (Φn))Φn)

(Φn,Φn)= E (Φn) . (10.55)

Similar to the proof of (10.22), we have

M−1∑

j=1

(φnj

)2 (φn+1j

)2≥

M−1∑

j=1

(φn+1j

)4. (10.56)

The required result (10.52) is a combination of (10.56), and (10.55). 2

1.10.8 Numerical results

Here we report the ground state solutions in BEC with different potentials by the methodBEFD. Due to the ground state solution φg(x) ≥ 0 for x ∈ Ω in non-rotating BEC [87],in our computations, the initial condition (10.3) is always chosen such that φ0(x) ≥ 0 anddecays to zero sufficiently fast as |x| → ∞. We choose an appropriately large interval,rectangle and box in 1d, 2d and 3d, respectively, to avoid that the homogeneous periodicboundary condition (10.35) introduce a significant (aliasing) error relative to the wholespace problem. To quantify the ground state solution φg(x), we define the radius meansquare

αrms = ‖αφg‖L2(Ω) =

√∫

Ωα2φ2

g(x) dx, α = x, y, or z. (10.57)

Example 1.1 Ground state solution of 1d BEC with harmonic oscillator potential

V (x) =x2

2, φ0(x) =

1

(π)1/4e−x

2/2, x ∈ R.

The CNGF (10.15)-(10.16) with d = 1 is solved on Ω = [−16, 16] with mesh size h =1/8 and time step k = 0.1 by using BEFD. The steady state solution is reached whenmax

∣∣Φn+1 − Φn∣∣ < ε = 10−6. Figure 1.1 shows the ground state solution φg(x) and energy

evolution for different β. Table 1.1 displays the values of φg(0), radius mean square xrms,energy E(φg) and chemical potential µg.

The results in Figure 1.1 and Table 1.1 agree very well with the ground state solutionsof BEC obtained by directly minimizing the energy functional [5].

Example 1.2 Ground state solution of BEC in 2d. Two cases are considered:


Table 1.1: Maximum value of the wave function φg(0), root mean square size xrms, energyE(φg) and ground state chemical potential µg verus the interaction coefficient β in 1d.

β φg(0) xrms E(φg) µg = µβ(φg)

0 0.7511 0.7071 0.5000 0.50003.1371 0.6463 0.8949 1.0441 1.527212.5484 0.5301 1.2435 2.2330 3.598631.371 0.4562 1.6378 3.9810 6.558762.742 0.4067 2.0423 6.2570 10.384156.855 0.3487 2.7630 11.464 19.083313.71 0.3107 3.4764 18.171 30.279627.42 0.2768 4.3757 28.825 48.0631254.8 0.2467 5.5073 45.743 76.312

a)0 2 4 6 8 10 12 14

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

Φ g(x

)

b)0 0.1 0.2 0.3 0.4 0.5 0.6

0.5

1

1.5

2

2.5

3

t

En

erg

y

β=12.5484, E(U) β=156.855, E(U)/10 β=627.4, E(U)/50 β=1254.8, E(U)/100

Figure 1.1: Ground state solution φg in Example 1.1. (a). For β =0, 3.1371, 12.5484, 31.371, 62.742, 156.855, 313.71, 627.42, 1254.8 (with decreasing peak).(b). Energy evolution for different β.

I. With a harmonic oscillator potential [5, 8, 44], i.e.

V (x, y) =1

2

(γ2xx

2 + γ2yy

2).

II. With a harmonic oscillator potential and a potential of a stirrer corresponding afar-blue detuned Gaussian laser beam [27] which is used to generate vortices in BEC [27],i.e.

V (x, y) =1

2

(γ2xx

2 + γ2yy

2)

+ w0e−δ((x−r0)2+y2).

The initial condition is chosen as

φ0(x, y) =(γxγy)

1/4

π1/2e−(γxx2+γyy2)/2.


(a).−5

0

5

−2

−1

0

1

2

0

0.05

xy

φ2 g

(b).

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

0

0.01

0.02

0.03

0.04

y

x

φ2 g

Figure 1.2: Ground state solutions φ2g in Example 1.2, case I (a), and case II (b).

For the case I, we choose γx = 1, γy = 4, w0 = δ = r0 = 0, β = 200 and solve theproblem by BEFD on Ω = [−8, 8] × [−4, 4] with mesh size hx = 1/8, hy = 1/16 and timestep k = 0.1. We get the following results from the ground state solution φg:

xrms = 2.2734, yrms = 0.6074, φ2g(0) = 0.0808, E(φg) = 11.1563, µg = 16.3377.

For case II, we choose γx = 1, γy = 1, w0 = 4, δ = r0 = 1, β = 200 and solve theproblem by TSSP on Ω = [−8, 8]2 with mesh size h = 1/8 and time step k = 0.001. Weget the following results from the ground state solution φg:

xrms = 1.6951, yrms = 1.7144, φ2g(0) = 0.034, E(φg) = 5.8507, µg = 8.3269.

In addition, Figure 1.2 shows surface plots of the ground state solution φg.

Example 1.3 Ground state solution of BEC in 3d. Two cases are considered:

I. With a harmonic oscillator potential [5, 8, 44], i.e.

V (x, y, z) =1

2

(γ2xx

2 + γ2yy

2 + γ2zz

2).

II. With a harmonic oscillator potential and a potential of a stirrer corresponding afar-blue detuned Gaussian laser beam [27] which is used to generate vortex in BEC [27],i.e.

V (x, y, z) =1

2

(γ2xx

2 + γ2yy

2 + γ2zz

2)

+ w0e−δ((x−r0)2+y2).

The initial condition is chosen as

φ0(x, y, z) =(γxγyγz)

1/4

π3/4e−(γxx2+γyy2+γzz2)/2.

For case I, we choose γx = 1, γy = 2, γz = 4, w0 = δ = r0 = 0, β = 200 and solvethe problem by TSSP on Ω = [−8, 8] × [−6, 6] × [−4, 4] with mesh size hx = 1

8 , hy = 332 ,

hz = 116 and time step k = 0.001. The ground state solution φg gives:

xrms = 1.67, yrms = 0.87, zrms = 0.49, φ2g(0) = 0.052, E(φg) = 8.33, µg = 11.03.


a).−5

0

5

−2

−1

0

1

20

0.02

0.04

xz

φ2 g

b). −5

0

5

−4

−2

0

2

40

0.01

0.02

0.03

xz

φ2 g

Figure 1.3: Ground state solutions φ2g(x, 0, z) in Example 1.3. (a). For case I. (b). For

case II.

For case II, we choose γx = 1, γy = 1, γz = 2, w0 = 4, δ = r0 = 1, β = 200 and solvethe problem by TSSP on Ω = [−8, 8]3 with mesh size h = 1

8 and time step k = 0.001. Theground state solution φg gives:

xrms = 1.37, yrms = 1.43, zrms = 0.70, φ2g(0) = 0.025, E(φg) = 5.27, µg = 6.71.

Furthermore, Figure 1.3 shows surface plots of the ground state solution φ2g(x, 0, z). BEFD

gives similar results with k = 0.1.

Example 1.4 2d central vortex states in BEC, i.e.

V (x, y) = V (r) =1

2

(m2

r2+ r2

), φ0(x, y) = φ0(r) =

1√πm!

rm e−r2/2, 0 ≤ r.

The CNGF (10.15)-(10.16) is solved in polar coordinate with Ω = [0, 8] with mesh sizeh = 1

64 and time step k = 0.1 by using BEFD. Figure 1.4a shows the ground state solutionφg(r) with β = 200 for different index of the central vortex m. Table 1.2 displays thevalues of φg(0), radius mean square rrms, energy E(φg) and chemical potential µg.

Table 1.2: Numerical results for 2d central vortex states in BEC.Index m φg(0) rrms E(φg) µg = µβ(φg)

1 0.0000 2.4086 5.8014 8.29672 0.0000 2.5258 6.3797 8.74133 0.0000 2.6605 7.0782 9.31604 0.0000 2.8015 7.8485 9.97725 0.0000 2.9438 8.6660 10.69946 0.0000 3.0848 9.5164 11.4664


(a).0 2 4 6

0

0.05

0.1

0.15

0.2

r

φ g(r)

(b).0 5 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

φ 1(x)

Figure 1.4: (a). 2d central vortex states φg(r) in Example 1.4. β = 200 form = 1 to 6 (withdecreasing peak). (b). First excited state solution φ1(x) (an odd function) in Example1.5. For β = 0, 3.1371, 12.5484, 31.371, 62.742, 156.855, 313.71, 627.42, 1254.8 (withdecreasing peak).

Example 1.5. The first excited state solution of BEC in 1d with a harmonic oscillatorpotential, i.e.

V (x) =x2

2, φ0(x) =

√2

(π)1/4x e−x

2/2, x ∈ R.

The CNGF (10.15)-(10.16) with d = 1 is solved on Ω = [−16, 16] with mesh size h = 1/64and time step k = 0.1 by using BEFD. Figure 1.4b shows the first excited state solutionφ1(x) for different β. Table 1.3 displays the radius mean square xrms = ‖xφ1‖L2(Ω),ground state and first excited state energies E(φg) and E(φ1), ratio E(φ1)/E(φg), chemicalpotentials µg = µβ(φg) and µ1 = µβ(φ1), ratio µ1/µg.

Table 1.3: Numerical results for the first excited state solution in 1d in Example 1.5.

β xrms E(φg) E(φ1)E(φ1)E(φg) µg µ1

µ1

µg

0 1.2247 0.500 1.500 3.000 0.500 1.500 3.0003.1371 1.3165 1.044 1.941 1.859 1.527 2.357 1.54412.5484 1.5441 2.233 3.037 1.360 3.598 4.344 1.20731.371 1.8642 3.981 4.743 1.192 6.558 7.279 1.11062.742 2.2259 6.257 6.999 1.119 10.38 11.089 1.068156.855 2.8973 11.46 12.191 1.063 19.08 19.784 1.037313.71 3.5847 18.17 18.889 1.040 30.28 30.969 1.023627.42 4.4657 28.82 29.539 1.025 48.06 48.733 1.0141254.8 5.5870 45.74 46.453 1.016 76.31 76.933 1.008

From the results in Table 1.3 and Figure 1.4b, we can see that the BEFD can be


applied directly to compute the first excited states in BEC. Furthermore, we have

limβ→+∞

E(φ1)

E(φg)= 1, lim

β→+∞

µ1

µg= 1.

These results are confirmed with the results in [5] where the ground and first excited statesare computed by directly minimizing the energy functional through the finite elementdiscretization.

1.11 Numerical methods for dynamics of NLSE

In this section we present time-splitting sine pseudospectral (TSSP) methods for the prob-lem (4.1), (4.2) with/without external driven field with homogeneous Dirichlet boundaryconditions. For the simplicity of notation we shall introduce the method for the case ofone space dimension (d = 1). Generalizations to d > 1 are straightforward for tensorproduct grids and the results remain valid without modifications. For d = 1, the problemwith an external driven field becomes

i∂tψ = −1

2∂xxψ + V (x)ψ +W (x, t)ψ + β|ψ|2ψ, a < x < b, t > 0, (11.1)

ψ(x, t = 0) = ψ0(x), a ≤ x ≤ b, ψ(a, t) = ψ(b, t) = 0, t ≥ 0; (11.2)

where W (x, t) is an external driven field. Typical external driven fields used in physicalliteratures include a far-blued detuned Gaussian laser beam stirrer [27]

W (x, t) = Ws(t) exp

[−(|x− xs(t)|2

ws/2

)], (11.3)

with Ws the height, ws the width, and xs(t) the position of the stirrer; or a Delta-kickedpotential [77]

W (x, t) = K cos(kx)∞∑

n=−∞

δ(t− nτ), (11.4)

with K the kick strength, k the wavenumber, τ the time interval between kicks, and δ(τ)is the Dirac delta function.

1.11.1 General high-order split-step method

As preparatory steps, we begin by introducing the general high-order split-step method[53] for a general evolution equation

i ∂tu = f(u) = A u+B u, (11.5)

where f(u) is a nonlinear operator and the splitting f(u) = Au+Bu can be quite arbitrary,in particular, A and B do not need to commute. For a given time step k = ∆t > 0, lettn = n k, n = 0, 1, 2, . . . and un be the approximation of u(tn). A second-order symplectictime integrator (cf. [111]) for (11.5) is as follows:

u(1) = e−ik A/2 un;

u(2) = e−ik B u(1);

un+1 = e−ik A/2 u(2). (11.6)

1.11. NUMERICAL METHODS FOR DYNAMICS OF NLSE 41

A fourth-order symplectic time integrator (cf. [120]) for (11.5) is as follows:

u(1) = e−i2w1k A un;

u(2) = e−i2w2k B u(1);

u(3) = e−i2w3k A u(2);

u(4) = e−i2w4k B u(3);

u(5) = e−i2w3k A u(4);

u(6) = e−i2w2k B u(5);

un+1 = e−i2w1k A u(6); (11.7)

where

w1 = 0.33780 17979 89914 40851, w2 = 0.67560 35959 79828 81702,

w3 = −0.08780 17979 89914 40851, w4 = −0.85120 71979 59657 63405.

1.11.2 Fourth-order TSSP for GPE without external driven field

We choose the spatial mesh size h = 4x > 0 with h = (b− a)/M for M an even positiveinteger, and let xj := a+ j h, j = 0, 1, · · · ,M . Let ψnj be the approximation of ψ(xj , tn)and ψn be the solution vector at time t = tn = nk with components ψnj .

We now rewrite the GPE (11.1) without external driven field, i.e. W (x, t) ≡ 0, in theform of (11.5) with

Aψ = V (x)ψ(x, t) + β|ψ(x, t)|2ψ(x, t), Bψ = −1

2∂xxψ(x, t). (11.8)

Thus, the key for an efficient implementation of (11.7) is to solve efficiently the followingtwo subproblems:

i ∂tψ(x, t) = Bψ = −1

2∂xxψ, (11.9)

andi ∂tψ(x, t) = V (x)ψ(x, t) + β|ψ(x, t)|2ψ(x, t). (11.10)

Equation (11.9) will be discretized in space by the sine pseudospectral method and inte-grated in time exactly. For t ∈ [tn, tn+1], the ODE (11.10) leaves |ψ| invariant in t [11, 10]and therefore becomes

iψt(x, t) = V (x)ψ(x, t) + β|ψ(x, tn)|2ψ(x, t) (11.11)

and thus can be integrated exactly.

From time t = tn to t = tn+1, we combine the splitting steps via the fourth-ordersplit-step method and obtain a fourth-order time-splitting sine-spectral (TSSP4) methodfor the GPE (10.33). The detailed method is given by

ψ(1)j = e−i2w1k(V (xj)+β|ψn

j |2) ψnj ,

ψ(2)j =

M−1∑

l=1

e−iw2kµ2l ψ

(1)l sin(µl(xj − a)),


ψ(3)j = e−i2w3k(V (xj)+β|ψ

(2)j |2) ψ

(2)j ,

ψ(4)j =

M−1∑

l=1

e−iw4kµ2l ψ

(3)l sin(µl(xj − a)), j = 1, 2, · · · ,M − 1,

ψ(5)j = e−i2w3k(V (xj)+β|ψ

(4)j |2) ψ

(4)j ,

ψ(6)j =

M−1∑

l=1

e−iw2kµ2l ψ

(5)l sin(µl(xj − a)),

ψn+1j = e−i2w1k(V (xj)+β|ψ

(6)j |2) ψ

(6)j , (11.12)

where Ul, the sine-transform coefficients of a complex vector U = (U0, U1, · · · , UM ) withU0 = UM = 0, are defined as

µl =πl

b− a, Ul =

2

M

M−1∑

j=1

Uj sin(µl(xj − a)), l = 1, 2, · · · ,M − 1, (11.13)

with

ψ0j = ψ(xj , 0) = ψ0(xj), j = 0, 1, 2, · · · ,M. (11.14)

Note that the only time discretization error of TSSP4 is the splitting error, which is fourthorder in k for any fixed mesh size h > 0.

This scheme is explicit, time reversible, just as the IVP for the GPE. Also, a mainadvantage of the time-splitting method is its time-transverse invariance, just as it holdsfor the GPE itself. If a constant α is added to the potential V1, then the discrete wavefunctions ψn+1

j obtained from TSSP4 get multiplied by the phase factor e−iα(n+1)k, whichleaves the discrete quadratic observables unchanged. This property does not hold for finitedifference schemes.

1.11.3 Second-order TSSP for GPE with external driven field

We now rewrite the GPE (11.1) with an external driven field

Aψ = −1

2∂xxψ(x, t), Bψ = V (x)ψ(x, t) +W (x, t)ψ(x, t) + β|ψ(x, t)|2ψ(x, t). (11.15)

Due to the external driven field could be vary complicated, e.g. it may be a Delta-function [77], here we only use a second-order split-step scheme in time discretization.More precisely, from time t = tn to t = tn+1, we proceed as follows:

ψ∗j =

M−1∑

l=1

e−ikµ2l /4 (ψn)l sin(µl(xj − a)), j = 1, 2, · · · ,M − 1,

ψ∗∗j = exp

[−ik(V (xj) + β|ψnj |2) − i

∫ tn+1

tnW (xj, t)dt

]ψ∗j ,

ψn+1j =

M−1∑

l=1

e−ikµ2l /4 (ψ∗∗)l sin(µl(xj − a)). (11.16)


Remark 1.11.1 If the integral in (11.16) could not be evaluated analytically, one can usenumerical quadrature to evaluate, e.g.

∫ tn+1

tnW (xj , t)dt ≈

k

6[W (xj, tn) + 4W (xj , tn + k/2) +W (xj, tn+1)] .

1.11.4 Stability

Let U = (U0, U1, · · · , UM )T with U0 = UM = 0, f(x) a homogeneous periodic function onthe interval [a, b], and let ‖ · ‖l2 be the usual discrete l2-norm on the interval (a, b), i.e.,

‖U‖l2 =

√√√√b− a

M

M−1∑

j=1

|Uj |2, ‖f‖l2 =

√√√√b− a

M

M−1∑

j=1

|f(xj)|2. (11.17)

For the stability of the second-order time-splitting spectral approximations TSSP2(11.16) and fourth-order scheme (11.12), we have the following lemma, which shows thatthe total charge is conserved.

Lemma 1.11.1 The second-order time-splitting sine pseudospectral scheme (11.16) andfourth-order scheme (11.12) are unconditionally stable. In fact, for every mesh size h > 0and time step k > 0,

‖ψn‖l2 = ‖ψ0‖l2 = ‖ψ0‖l2 , n = 1, 2, · · · (11.18)

Proof: For the scheme TSSP2 (11.16), noting (10.42) and (11.17), one has

1

b− a‖ψn+1‖2

l2 =1

M

M−1∑

j=1

∣∣∣ψn+1j

∣∣∣2

=1

M

M−1∑

j=1

∣∣∣∣∣M−1∑

l=1

e−ikµ2l /4 (ψ∗∗)l sin(µl(xj − a))

∣∣∣∣∣

2

=1

M

M−1∑

j=1

∣∣∣∣∣M−1∑

l=1

e−ikµ2l/4 (ψ∗∗)l sin(jlπ/M)

∣∣∣∣∣

2

=1

2

M−1∑

l=1

∣∣∣e−ikµ2l /4 (ψ∗∗)l

∣∣∣2

=1

2

M−1∑

l=1

∣∣∣(ψ∗∗)l

∣∣∣2

=1

2

M−1∑

l=1

∣∣∣∣∣∣2

M

M−1∑

j=1

ψ∗∗j sin(µl(xj − a))

∣∣∣∣∣∣

2

=1

2

M−1∑

l=1

∣∣∣∣∣∣2

M

M−1∑

j=1

ψ∗∗j sin(ljπ/M)

∣∣∣∣∣∣

2

=1

M

M−1∑

j=1

∣∣∣ψ∗∗j

∣∣∣2

=1

M

M−1∑

j=1

∣∣∣∣exp

[−ik(V (xj) + β|ψnj |2) − i

∫ tn+1

tnW (xj, t)dt

]ψ∗j

∣∣∣∣2

=1

M

M−1∑

j=1

∣∣∣ψ∗j

∣∣∣2

=1

M

M−1∑

j=1

∣∣∣ψnj∣∣∣2

=1

b− a‖ψn‖2

l2 . (11.19)

Here we used the identities

M−1∑

j=1

sin(ljπ/M) sin(kjπ/M) =

0, k = l,M/2, k 6= l.

(11.20)


For the scheme TSSP4 (11.12), using (10.42), (11.17) and (11.20), we get similarly

1

b− a‖ψn+1‖2

l2 =1

M

M−1∑

j=0

∣∣∣ψn+1j

∣∣∣2

=1

M

M−1∑

j=0

∣∣∣∣e−i2w1k(V (xj)+β|ψ

(6)j |2) ψ

(6)j

∣∣∣∣2

=1

M

M−1∑

j=0

∣∣∣ψ(6)j

∣∣∣2

=1

M

M−1∑

j=0

∣∣∣∣∣M−1∑

l=1

e−iw2kµ2l ψ

(5)l sin(µl(xj − a))

∣∣∣∣∣

2

=1

M

M−1∑

j=0

∣∣∣∣∣M−1∑

l=1

e−iw2kµ2l ψ

(5)l sin(ljπ/M)

∣∣∣∣∣

2

=1

2

M−1∑

l=1

∣∣∣e−iw2kµ2l ψ

(5)l

∣∣∣2

=1

2

M−1∑

l=1

∣∣∣ψ(5)l

∣∣∣2

=1

2

M−1∑

l=1

∣∣∣∣∣∣2

M

M−1∑

j=1

ψ(5)j sin(µl(xj − a))

∣∣∣∣∣∣

2

=1

2

M−1∑

l=1

∣∣∣∣∣∣2

M

M−1∑

j=1

ψ(5)j sin(jlπ/M)

∣∣∣∣∣∣

2

=1

M

M−1∑

j=1

∣∣∣ψ(5)j

∣∣∣2

=1

M

M−1∑

j=1

∣∣∣ψ(4)j

∣∣∣2

=1

M

M−1∑

j=1

∣∣∣ψ(3)j

∣∣∣2

=1

M

M−1∑

j=1

∣∣∣ψ(2)j

∣∣∣2

=1

M

M−1∑

j=1

∣∣∣ψ(1)j

∣∣∣2

=1

M

M−1∑

j=1

∣∣∣ψnj∣∣∣2

=1

b− a‖ψn‖2

l2 . (11.21)

Thus the equality (11.18) can be obtained from (11.19) for the scheme TSSP2 and (11.21)for the scheme TSSP4 by induction. 2

1.11.5 Crank-Nicolson finite difference method (CNFD)

Another scheme used to disretize the NLSE (11.1) is the Crank-Nicolson finite differencemethod (CNFD). In this method one uses the Crank-Nicolson scheme for time derivativeand the second order central difference scheme for spatial derivative. The detailed methodis:

iψn+1j − ψnj

k= − 1

4h2

(ψn+1j+1 − 2ψn+1

j + ψn+1j−1 + ψnj+1 − 2ψnj + ψnj−1

)

+V (xj)

2

(ψn+1j + ψnj

)+β

2

(|ψn+1j |2 + |ψnj |2

) (ψn+1j + ψnj

), (11.22)

j = 1, 2, · · · ,M − 1, n = 0, 1, · · · ,ψn+1

0 = ψn+1M = 0, n = 0, 1, · · · ,

ψ0j = ψ0(xj), j = 0, 1, 2, · · · ,M.


1.11.6 Numerical results

In this subsection we present numerical results to confirm spectral accuracy in space andfourth order accuracy in time of the numerical method (11.12), and then apply it to studytime-evolution of condensate width in 1D, 2D and 3D.

Example 1.6 1d Gross-Pitaevskii equation, i.e. in (4.1) we choose d = 1 and γx = 1.The initial condition is taken as

ψ0(x) =1

π1/4e−x

2/2, x ∈ R.

We solve on the interval [−32, 32], i.e. a = −32 and b = 32 with homogeneous Dirichletboundary condition (11.2). We compute a numerical solution by using TSSP4 with a veryfine mesh, e.g. h = 1

128 , and a very small time step, e.g. k = 0.0001, as the ‘exact’ solutionψ. Let ψh,k denote the numerical solution under mesh size h and time step k.

First we test the spectral accuracy of TSSP4 in space. In order to do so, for eachfixed β1, we solve the problem with different mesh size h but a very small time step, e.g.k = 0.0001, such that the truncation error from time discretization is negligible comparingto that from space discretization. Table 1.4 shows the errors ‖ψ(t) − ψh,k(t)‖l2 at t = 2.0with k = 0.0001 for different β1 and h.

mesh h = 1 h = 12 h = 1

4 h = 18 h = 1

16β1 = 10 0.2745 1.081E-2 1.805E-6 3.461E-11

β1 = 20√

2 1.495 0.1657 7.379E-4 7.588E-10β1 = 80 1.603 1.637 6.836E-2 3.184E-5 3.47E-11

Table 1.4: Spatial error analysis: Error ‖ψ(t) − ψh,k(t)‖l2 at t = 2.0 with k = 0.0001 inExample 1.6.

Then we test the fourth-order accuracy of TSSP4 in time. In order to do so, for eachfixed β1, we solve the problem with different time step k but a very fine mesh, e.g. h = 1

64 ,such that the truncation error from space discretization is negligible comparing to thatfrom time discretization. Table 1.5 shows the errors ‖ψ(t) − ψh,k(t)‖l2 at t = 2.0 withh = 1

64 for different β1 and k.

As shown in Tables 1.4&5, spectral order accuracy for spatial derivatives and fourth-order accuracy for time derivative of TSSP4 are demonstrated numerically for 1d GPE,respectively. Another issue is how to choose mesh size h and time step k in the strongrepulsive interaction regime or semiclassical regime, i.e. βd 1, in order to get “correct”physical observables. In fact, after a rescaling in (4.1) under the normalization (4.3):

x → ε−1/2x and ψ → εd/4ψ with ε = β−2/(d+2)d , then the GPE (4.1) can be rewritten as

iε ∂tψ(x, t) = −ε2

2∇2ψ + Vd(x)ψ + |ψ|2ψ, x ∈ Rd. (11.23)


time step k = 120 k = 1

40 k = 180 k = 1

160 k = 1320 k = 1

640δ = 10.0 1.261E-4 8.834E-6 5.712E-7 3.602E-8 2.254E-9 1.422E-10

δ = 20√

2 1.426E-3 9.715E-5 6.367E-6 4.034E-7 2.529E-8 1.580E-9δ = 80 4.375E-2 1.693E-3 8.982E-5 5.852E-6 3.706E-7 2.323E-8

Table 1.5: Temporal error analysis: ‖ψ(t) − ψh,k(t)‖l2 at t = 2.0 with h = 164 in Example

1.6.

a)0 2 4 6 8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t

σ x (or

|ψ(0

,t)|2 )

b)

0

1

2

3

4

5

6 0

0.2

0.4

0.6

0.8

1

0

0.5

1

tx

|ψ|2

Figure 1.5: Numerical results for example 1.7: a) Condensate width σx ( ‘—’ ) andcentral density |ψ(0, t)|2 ( ‘- - - ’ ). b) Evolution of the density function |ψ|2.

As demonstrated in [10, 11], the meshing strategy to capture ‘correct’ physical observablesfor this this problem is

h = O(ε), k = O(ε).

Thus the admissible meshing strategy for the GPE with strong repulsive interaction is

h = O(ε) = O(1/β

2/(d+2)d

), k = O(ε) = O

(1/β

2/(d+2)d

), d = 1, 2, 3. (11.24)

Example 1.7 1d Gross-Pitaevskii equation, i.e. in (4.1) we choose d = 1. The initialcondition is taken as the ground-state solution of (4.1) under d = 1 with γx = 1 andβ1 = 20.0 [6, 5], i.e. initially the condensate is assumed to be in its ground state. Whent = 0, we double the trap frequency by setting γx = 2.

We solve this problem on the interval [−12, 12] under mesh size h = 364 and time

step k = 0.005 with homogeneous Dirichlet boundary condition. Figure 1.5 plots thecondensate width and central density |ψ(0, t)|2 as functions of time, as well as evolutionof the density |ψ|2 in space-time. One can see from this figure that the sudden change inthe trap potential leads to oscillations in the condensate width and the peak value of thewave function. Note that the condensate width contracts in an oscillatory way (cf. Fig.1.5a), which agrees with the analytical results in (4.43).


a)0 5 10 15

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

t

σx

σy

b) −4

−2

0

2

4

−2

−1

0

1

2

0

0.1

0.2

0.3

x

t=5.4

y

|ψ|2

Figure 1.6: Numerical results for example 1.8: a) Condensate width. b) Surface plotof the density |ψ|2 at t = 5.4.

Example 1.8 2d Gross-Pitaevskii equation, i.e. in (4.1) we choose d = 2. The initialcondition is taken as the ground-state solution of (4.1) under d = 2 with γx = 1, γy = 2and β2 = 20.0 [6, 5], i.e. initially the condensate is assumed to be in its ground state.When at t = 0, we double the trap frequency by setting γx = 2 and γy = 4.

We solve this problem on [−8, 8]2 under mesh size h = 132 and time step k = 0.005

with homogeneous Dirichlet boundary condition. Figure 1.6 shows the condensate widthsσx and σy as functions of time and the surface of the density |ψ|2 at time t = 5.4. Figure1.7 the contour plots of the density |ψ|2 at different times. Again, the sudden change inthe trap potential leads to oscillations in the condensate width. Due to γy = 2γx, theoscillation frequency of σy is roughly double that of σx and the amplitudes of σx are largerthan those of σy in general (cf. Fig. 1.6a). Again this agrees with the analytical resultsin (4.43).

Example 1.9 3d Gross-Pitaevskii equation, i.e. in (4.1) we choose d = 3. We presentcomputations for two cases:

Case I. Intermediate ratio between trap frequencies along different axis (data for 87Rbused in JILA [3]). The initial condition is taken as the ground-state solution of (4.1)under d = 3 with γx = γy = 1, γz = 4 and β3 = 37.62 [6, 5]. When at t = 0, we four timesthe trap frequency by setting γx = γy = 4 and γz = 16.

Case II. High ratio between trap frequencies along different axis (data for 23Na used inMIT (group of Ketterle) [38]). The initial condition is taken as the ground-state solutionof (4.1) under d = 3 with γx = γy = 360

3.5 , γz = 1 and β3 = 3.083 [6, 5]. When at t = 0, wedouble the trap frequency by setting γx = γy = 720

3.5 and γz = 2.

For case I, we solve the problem on [−6, 6] × [−6, 6] × [−3, 3] under mesh size hx =hy = 3

32 and hz = 364 , and time step k = 0.0025 with homogeneous Dirichlet boundary

condition. For case II, we solve the problem on [−0.5, 0.5]×[−0.5, 0.5]×[−8, 8] under meshsize hx = hy = 1

128 and hz = 18 , and time step k = 0.0005 with homogeneous Dirichlet

boundary condition along their boundaries.


a).−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4

x

y

t=0

b)−2 −1 0 1 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

y

t=0.9

c).−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4

x

y

t=1.8

d)−2 −1 0 1 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

y

t=2.7

e).−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4

x

y

t=3.6

f)−1.5 −1 −0.5 0 0.5 1 1.5

−1.5

−1

−0.5

0

0.5

1

1.5

x

y

t=4.5

Figure 1.7: Contour plots of the density |ψ|2 at different times in Example 1.8. a). t = 0,b). t = 0.9, c). t = 1.8, d). t = 2.7, e). t = 3.6, f). t = 4.5.

Figure 1.8 shows the condensate widths σx = σy and σz as functions of time, as wellas the surface of the density in xz-plane |ψ(x, 0, z, t)|2 . Similar phenomena in case I in3d is observed as those in Example 1.8 which is in 2d (cf. Fig. 1.6a). The ratio betweenthe condensate widths increases with increasing the ratio between trap frequencies alongdifferent axis, i.e. it becomes more difficult to excite oscillations for large trap frequencies.In case II, the curves of the condensate widths are very well separated. This behavior isone of the basic assumptions allowing the reduction of GPE to 2d and 1d in the cases oneor two of the trap frequencies are much larger than the others [85, 5, 8].


a)0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

t

σz

σx=σ

y

b) −4

−2

0

2

4

−1.5

−1

−0.5

0

0.5

1

1.5

0

0.05

0.1

xz|ψ

|2

c)0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t

σz

σx=σ

y

d)−0.2

−0.1

0

0.1

0.2

−4

−2

0

2

4

0

10

20

xz

|ψ|2

Figure 1.8: Numerical results for example 1.9: Left column: Condensate width; rightcolumn: Surface plot of the density in xz-plane, |ψ(x, 0, z, t)|2 . Case I: a) and b) att = 1.64. Case II: c) and d) at t = 4.5.

Chapter 2

The Zakharov system

2.1 Introduction

The Zakharov system (ZS) was derived by Zakharov [121] in 1972 for governing the coupleddynamics of the electric-field amplitude and the low-frequency density fluctuations of ions.Then it has become commonly accepted that ZS is a general model to govern interaction ofdispersive wave and nondispersive (acoustic) wave. It has important applications in plasmaphysics (interaction between Langmuir and ion acoustic waves [121, 101]), in the theory ofmolecular chains (interaction of the intramolecular vibrations forming Davydov solitonswith the acoustic disturbances in the chain [39]), in hydrodynamics (interaction betweenshort-wave and long-wave gravitational disturbances in the atmosphere [110, 40]), and soon. In three spatial dimensions, ZS was also derived to model the collapse of caverns [121].Later, the standard ZS was extended to generalized Zakharov system (GZS) [72, 73], vectorZakharov system (VZS) [113] and vector Zakharov system for multi-component plasma(VZSM) [72, 73].

In this chapter, we first review derivation of VZS from the two-fluid model [113] forion-electron dynamics in plasma physics and generalize VZS to VZSM. Then we reduceVZSM to generalized vector Zakharov system (GVZS), GVZS to GZS or vector nonlinearSchrodinger (VNLS) equations, and GZS to NLS equation, as well as generalize GZS andGVZS with a linear damping term to arrest blowup. Conservation laws of the systemsand well-posedness of GZS are presented, and plane wave, soliton wave and periodic wavesolutions of GZS are reviewed. Furthermore, we present a time-splitting spectral (TSSP)method and a Crank-Nicolson finite difference (CNFD) method to discretize GZS. Numer-ical results are reported.

2.2 Derivation of the vector Zakharov system

In this section, we derive VZS from the two-fluid model [113] for ion-electron dynamicsin plasma physics. Here we will use a more formal approach based on the multiple-scalemodulation analysis. Following from [113], we will consider a plasma as two interpene-trating fluids, an electron fluid and an ion fluid, and denote the mass, number density(number of particles per unit volume) and velocity of the electrons (respectively of theions), by me, Ne(x, t) and ve(x, t) (respectively mi, Ni(x, t) and vi(x, t)). The continuity

51

52 CHAPTER 2. THE ZAKHAROV SYSTEM

equations for these fluids read

∂tNe + ∇ · (Neve) = 0, (2.1)

∂tNi + ∇ · (Nivi) = 0, x ∈ R3, t > 0 (2.2)

and the momentum equations read

meNe(∂tve + ve · ∇ve) = −∇pe − eNe

(E +

1

cve × B

), (2.3)

miNi(∂tvi + vi · ∇vi) = −∇pi + eNi

(E +

1

cvi × B

), (2.4)

where −e and e represent the charge of the electron and the ions assumed to reduceto protons, respectively; pe and pi are the pressure. For small fluctuations, we write∇pe = γe Te∇Ne and ∇pi = γi Ti∇Ni, where γe and γi denote the specific heat ratios ofthe electrons and the ions and Te and Ti their respective temperatures in energy units.The electric field E and magnetic field B are provided by the Maxwell equations

−1

c∂tE + ∇× B =

4π

cj, ∇ · E = 4πρ, (2.5)

1

c∂tB + ∇× E = 0, ∇ · B = 0, (2.6)

where ρ = −e(Ne − Ni) and j = −e(Neve − Nivi) are the densities of total charge andtotal current, respectively.

Equations (2.5) and (2.6) yield

1

c2∂ttE + ∇× (∇× E) +

4π

c2∂tj = 0, (2.7)

and using equations (2.1)-(2.4), we have

∂tj = e(∇ · (Neve)ve +Neve · ∇ve + 1me

∇pe + eNeme

(E + 1cve × B))

−e(∇ · (Nivi)vi +Nivi · ∇vi +1mi

∇pi + eNimi

(E + 1cvi × B)). (2.8)

In order to get VZS from the two-fluid model just mentioned, as in [113], we considera long-wavelength small-amplitude Langmuir oscillation of the form

E =ε

2(E(X, T )e−iωet + c.c.) + ε2E(X, T ) + · · · , (2.9)

where the complex amplitude E depends on the slow variables X = εx and T = ε2t, thenotation c.c. stands for the complex conjugate and E(X, T ) denotes the mean complexamplitude. It induces fluctuations for the density and velocity of the electrons and of theions whose dynamical time will be seen to be τ = εt, thus shorter than T . We write

Ne = N0 +ε2

2(Ne(X, τ)e

−iωet + c.c.) + ε2Ne(X, τ) + · · · , (2.10)

Ni = N0 +ε2

2(Ni(X, τ)e

−iωet + c.c.) + ε2Ni(X, τ) + · · · , (2.11)

ve =ε

2(ve(X, τ)e

−iωet + c.c.) + ε2ve(X, τ) + · · · , (2.12)

vi =ε

2(vi(X, τ)e

−iωet + c.c.) + ε2vi(X, τ) + · · · , (2.13)

2.2. DERIVATION OF THE VECTOR ZAKHAROV SYSTEM 53

where N0 is the unperturbed plasma density.

From the momentum equation (2.3), considering the leading order and noting that themagnetic field B is of order ε2, we can easily get

meNe

(iωeve

ε

2e−iωet

)= eNe

(Eε

2e−iωet

),

thus the amplitude of the electron velocity oscillations is given by

ve = − ie

meωeE. (2.14)

Neglecting the velocity oscillations of the ions due to their large mass, we take

vi = 0. (2.15)

Applying (2.14) and (2.15) into the continuity equations (2.1) and (2.2), at the order ofε2, we have

−iωeNeε2

2e−iωet +N0∇ · ve

ε2

2e−iωet = 0,

thus the density fluctuations are obtained as

Ne = −iN0

ωe∇ · ve = − eN0

meω2e

∇ ·E, (2.16)

Ni = 0. (2.17)

At leading order, the equation for the electric field (2.7) with j = −e(Neve − Nivi)becomes

− 1

c2ω2eE

ε

2e−iωet +

4π

c2iωeeN0ve

ε

2e−iωet = 0,

from which, with (2.14), we finally get the electron plasma frequency

ωe =

√4πe2N0

me. (2.18)

At the order of ε3, if no large-scale magnetic field is generated, then the equation (2.7)with (2.8) implies that

−2iωec2∂TE

ε3

2e−iωet + ∇× (∇× E)

ε3

2e−iωet

−4πe2N0γeTec2m2

eω2e

∇(∇ · E)ε3

2e−iωet +

4πe2NeE

c2me

ε3

2e−iωet = 0,

and thus

−2iωec2∂TE + ∇× (∇× E) − γeTe

mec2∇(∇ ·E) +

4πe2

c2meNeE = 0, (2.19)


where, resulting from (2.15) and (2.17), the contribution of the ions is negligible.

We rewrite the amplitude equation (2.19) as

i∂TE − c2

2ωe∇× (∇× E) +

3v2e

2ωe∇(∇ · E) =

ωe2

Ne

N0E, (2.20)

where the electron thermal velocity ve is defined by

ve =

√Teme

(2.21)

and γe is taken to be 3 [113].

It is seen from (2.3), (2.4) and (2.15) that the mean electron velocity ve and the meanion velocity vi satisfy

me

(∂τ ve +

1

4(ve · ∇v∗

e + v∗e · ∇ve)

)= −γeTe

N0∇Ne − eE , (2.22)

mi∂τ vi = −γiTiN0

∇Ni + eE , (2.23)

where

1

4(ve · ∇v∗

e + v∗e · ∇ve) =

e2

4m2eω

2e

∇|E|2, (2.24)

and ve denotes the conjugate of ve and me∂τ ve is negligible because of the small mass ofthe electron. Furthermore, E denotes the leading contribution (of order ε3) of the meanelectron field. We thus replace (2.22) by

e2

4meω2e

∇|E|2 = −γeTeN0

∇Ne − eE . (2.25)

The system is closed by using the quasi-neutrality of the plasma in the form

Ne = Ni, (2.26)

ve = vi, (2.27)

which we denote by N and v, respectively. Then from the continuity equations, one gets

∂τN +N0∇ · v = 0. (2.28)

Adding (2.25) to (2.23) and noting (2.28), we have

∂τv = − c2sN0

∇N − 1

16πmiN0∇|E|2, (2.29)

with the speed of sound cs,

c2s = ηTemi, η =

γeTe + γiTiTe

. (2.30)

2.3. GENERALIZATION AND SIMPLIFICATION 55

Finally, we obtain the VZS [113] from equations (2.20), (2.28) and (2.29) as

i∂TE − c2

2ωe∇× (∇× E) +

3v2e

2ωe∇(∇ ·E) =

ωe2

N

N0E, (2.31)

ε2∂TTN − c2s 4N =1

16πmi4 |E|2. (2.32)

This VZS governs the coupled dynamics of the electric-field amplitude and of the low-frequency density fluctuations of the ions and describes the dynamics of the complexenvelope of the electric field oscillations near the electron plasma frequency and the slowvariations of the density perturbations.

In order to obtain a dimensionless form of the system (2.31)-(2.32), we define thenormalized variables

t′ =2η

3µm ωe T, x′ =

2

3(ηµm)1/2

X

ζd, (2.33)

N ′ =3

4η

1

µm

N

N0, E′ =

1

η

1

µ1/2m

(3

64πN0Te

)1/2

E. (2.34)

with

ζd =

√Te

4πe2N0, µm =

me

mi, (2.35)

where ζd is the Debye length and µm is the ratio of the electron to the ion masses. Thendefining

a =c2

3v2e

=c2

3ω2eζ

2d

(2.36)

and plugging (2.33)-(2.34) into (2.31)-(2.32), and then removing all primes, we get thefollowing dimensionless vector Zakharov system in three dimension

i∂tE − a∇× (∇× E) + ∇(∇ · E) = N E, (2.37)

ε2∂ttN −4N = 4|E|2, x ∈ R3, t > 0. (2.38)

In fact, the equation (2.37) is equalivent to

i∂tE + a4 E + (1 − a)∇(∇ ·E) = N E. (2.39)

2.3 Generalization and simplification

The VZS (2.37), (2.38) can be easily generalized to a physical situation when the dispersivewaves interact with M different acoustic modes, e.g. in a multi-component plasma, whichmay be described by the following VZSM [113, 72, 73]:

i ∂tE + a 4 E + (1 − a) ∇(∇ · E) − EM∑

J=1

NJ = 0, x ∈ Rd, t > 0, (3.1)

ε2J ∂ttNJ −4NJ + νJ 4 |E|2 = 0, J = 1, · · · ,M; (3.2)


where the real unknown function NJ is the Jth-component deviation of the ion densityfrom its equilibrium value, εJ > 0 is a parameter inversely proportional to the acousticspeed of the Jth-component, and νJ are real constants.

The VZSM (3.1), (3.2) is time reversible and time transverse invariant, and preservesthe following three conserved quantities. They are the wave energy

DV ZSM =

∫

Rd|E(x, t)|2 dx, (3.3)

the momentum

PV ZSM =

∫

Rd

i

2

d∑

j=1

(Ej ∇E∗

j − E∗j ∇Ej

)−

M∑

J=1

ε2JνJNJVJ

dx (3.4)

and the Hamiltonian

HV ZSM =

∫

Rd

[a ‖∇E‖2

l2 + (1 − a)|∇ · E|2 +M∑

J=1

NJ |E|2

−1

2

M∑

J=1

(ε2JνJ

|VJ |2 +1

νJN2J

)]dx; (3.5)

where the flux vector VJ = ((vJ)1, · · ·, (vJ)d)T for the Jth-component is introduced

through the equations

∂tNJ = −∇ · VJ , ∂tVJ = − 1

ε2J∇(NJ − νJ |E|2), J = 1, · · · ,M. (3.6)

2.3.1 Reduction from VZSM to GVZS

In the VZSM (3.1)-(3.2), if we choose M = ∈, and assume that 1/ε22 1/ε21, i.e. theacoustic speed of the second component is much faster than the first component, thenformally the fast nondispersive component N2 can be excluded by means of the relation

N2 = ν2 |E|2 + ε22 4−1 ∂ttN2 ≈ ν2 |E|2 +O(ε22), when ε2 → 0. (3.7)

Plugging (3.7) into (3.1), then the VZSM (3.1), (3.2) is reduced to GVZS with N = N1,ν = ν1, ε = ε1, λ = −ν2 and α = 1:

i ∂tE + a 4 E + (1 − a) ∇(∇ · E) − α N E + λ |E|2E = 0, (3.8)

ε2∂ttN −4N + ν 4 |E|2 = 0, x ∈ Rd, t > 0. (3.9)

The GVZS (3.8), (3.9) is time reversible, time transverse invariant and preserves thefollowing three conserved quantities, i.e. the wave energy, momentum and Hamiltonian:

DGV ZS =

∫

Rd|E(x, t)|2 dx, (3.10)

PGV ZS =

∫

Rd

i

2

d∑

j=1

(Ej ∇E∗

j − E∗j ∇Ej

)− αε2

νNV

dx, (3.11)

HGV ZS =

∫

Rd

[a ‖∇E‖2

l2 + (1 − a)|∇ ·E|2 + αN |E|2 − λ

2|E|4

− α

2νN2 − αε2

2ν|V|2

]dx; (3.12)


where the flux vector V = (v1, · · · , vd)T is introduced through the equations

∂tN = −∇ · V, ∂tV = − 1

ε2∇(N − ν|E|2). (3.13)

In the case of M = ∈, ν = ν1 and ε = ε1, N = N1 and V = V1 in (3.4) and (3.5),and λ = −ν2, α = 1 in (3.11), (3.12), letting ε2 → 0 and noting (3.7), we get formallyquadratic convergence rate of the momentum and Hamiltonian from VZSM to GVZS inthe ‘subsonic limit’ regime of the second component, i.e., 0 < ε2 1:

PV ZSM =

∫

Rd

i

2

d∑

j=1

(Ej ∇E∗

j − E∗j ∇Ej

)− ε21ν1N1V

dx − ε22

ν2

∫

RdN2V2 dx

≈ PGV ZS +O(ε22), (3.14)

HV ZSM =

∫

Rd

[a ‖∇E‖2

l2 + (1 − a)|∇ ·E|2 +N1|E|2 − 1

2ν1N2

1 − ε212ν1

|V1|2]dx

+

∫

Rd

[N2|E|2 − 1

2ν2N2

2 − ε222ν2

|V2|2]dx

≈ HGV ZS +O(ε22). (3.15)

Choosing a = 1, α = 1, ν = −1 and λ = 0 in the GVZS (3.8)-(3.9), it collapses to thestandard VZS [113]

i ∂tE + 4E −NE = 0, x ∈ Rd, t > 0, (3.16)

ε2 ∂ttN −4N −4|E|2 = 0. (3.17)

2.3.2 Reduction from GVZS to GZS

In the case when E2 = · · · = Ed = 0 and a = 1 in the GVZS (3.8), (3.9), it reduces to thescalar GZS [113, 15],

i ∂tE + 4E − α N E + λ |E|2E = 0, x ∈ Rd, t > 0, (3.18)

ε2 ∂ttN −4N + ν 4 |E|2 = 0. (3.19)

The GZS (3.18), (3.19) is time reversible, time transverse invariant and conserved thefollowing wave energy, momentum and Hamiltonian:

DGZS =

∫

Rd|E(x, t)|2 dx, (3.20)

PGZS =

∫

Rd

[i

2(E∇E∗ − E∗∇E) − ε2α

νNV

]dx, (3.21)

HGZS =

∫

Rd

[|∇E|2 + αN |E|2 − λ

2|E|4 − α

2νN2 − αε2

2ν|V|2

]dx; (3.22)

where the flux vector V = (v1, · · · , vd)T is introduced through the equations

Nt = −∇ · V, Vt = − 1

ε2∇(N − ν|E|2). (3.23)


Choosing α = 1, ν = −1, ε = 1 and λ = 0 in the GZS (3.18)-(3.19), it collapses to thestandard ZS [113, 15, 121]. When λ 6= 0, a cubic nonlinear term is added to the standardZS.

Proof of the conservation laws in GZS: Multiplying (3.18) by E, the conjugate of E,we get

iEtE∗ + E∗ 4 E − αN |E|2 + λ |E|4 = 0. (3.24)

Then calculating the conjugate of (3.24) and multiplying it by E, one finds

−iE∗t E + E 4 E∗ − αN |E|2 + λ |E|4 = 0. (3.25)

Subtracting (3.25) from (3.24) and then multiplying both sides by −i, one gets

EtE∗ + E∗

tE + i(E 4 E∗ − E∗ 4 E) = 0. (3.26)

Integrating over Rd, integration by parts, (3.26) leads to the conservation of the waveenergy

d

dtDGZS =

d

dt

∫

Rd|E(x, t)|2 dx = 0.

From (3.21), noting (3.23), (3.18), one has the conservation of the momentum

d

dtPGZS =

i

2

∫

Rd(Et∇E∗ + E∇E∗

t − E∗∇Et − E∗t∇E) dx − ε2α

ν

∫

Rd(NtV +NVt) dx

= i

∫

Rd(Et∇E∗ − E∗

t∇E) dx − ε2α

ν

∫

Rd(NtV +NVt) dx

= i

∫

Rd∇E∗(i4 E − iαNE + iλ|E|2E)dx − ε2α

ν

∫

Rd(NtV +NVt) dx

−i∫

Rd∇E(−i4 E∗ + iαNE∗ − iλ|E|2E∗) dx

= α

∫

RdN∇|E|2 dx +

ε2α

ν

∫

RdV∇ ·V dx +

α

ν

∫

Rd∇(N − ν|E|2)N dx

= 0.

Noting (3.23), (3.19) and multiplying (3.18) by E∗t , the conjugate of Et, we write it

T =

∫

Rd

[i|Et|2 +E∗

t 4 E − αNEE∗t + λ|E|2EE∗

t

]dx = 0. (3.27)

Then the real part of T is

0 = Re(T ) = Re

∫

Rd

[E∗t 4 E − αNEE∗

t + λ|E|2EE∗t

]dx

= Re

∫

Rd

[−∇E∇E∗

t −α

2(N |E|2)t +

α

2Nt|E|2 +

λ

4(|E|4)t

]dx


= −1

2

∫

Rd

[(|∇E|2 + αN |E|2 − λ

2|E|4

)

t+α

2Nt|E|2

]dx

= −1

2

∫

Rd

[(|∇E|2 + αN |E|2 − λ

2|E|4

)

t− α

2|E|2∇ · V

]dx

= −1

2

∫

Rd

[(|∇E|2 + αN |E|2 − λ

2|E|4

)

t+α

2∇|E|2 · V

]dx

= −1

2

∫

Rd

[(|∇E|2 + αN |E|2 − λ

2|E|4

)

t+α

2(ε2

νVt +

1

ν∇N) ·V

]dx

= −1

2

∫

Rd

(|∇E|2 + αN |E|2 − λ

2|E|4

)dx

+αε2

2ν

∫

Rd

1

2(|V|2)t dx − α

2ν

∫

RdN∇ ·V dx

= −1

2

∫

Rd∂t

(|∇E|2 + αN |E|2 − λ

2|E|4

)dx

+αε2

2ν

∫

Rd

1

2(|V|2)t dx +

α

2ν

∫

RdN∂tN dx

= −1

2

∫

Rd∂t

(|∇E|2 + αN |E|2 − λ

2|E|4 − α

2νN2 − αε2

2ν|V|2)

)dx,

which implies the conservation of Hamiltonian

d

dtHGZS = 0.

2.3.3 Reduction from GVZS to VNLS

In the “subsonic limit”, i.e. ε → 0 in GVZS (3.8), (3.9), which corresponds to that thedensity fluctuations are assumed to follow adiabatically the modulation of the Langmuirwave, it collapses to the VNLS equation. In fact, letting ε→ 0 in (3.9), we get formally

N = ν |E|2 + ε2 4−1 ∂ttN = ν |E|2 +O(ε2), when ε→ 0. (3.28)

Plugging (3.28) into (3.8), we obtain formally the VNLS:

i ∂tE + a 4 E + (1 − a) ∇(∇ ·E) + (λ− αν)|E|2E = 0, x ∈ Rd, t > 0. (3.29)

The VNLS (3.29) is time reversible, time transverse invariant and preserves the followingwave energy, momentum and Hamiltonian:

DV NLS =

∫

Rd|E(x, t)|2 dx, (3.30)

PV NLS =

∫

Rd

i

2

d∑

j=1

(Ej ∇E∗

j − E∗j ∇Ej

)dx, (3.31)

HV NLS =

∫

Rd

[a ‖∇E‖2

l2 + (1 − a)|∇ · E|2 +αν − λ

2|E|4

]dx. (3.32)

Letting ε → 0 in (3.11), (3.12), noting (3.28), we get formally quadratic convergencerate of the momentum and Hamiltonian from GVZS to VNLS in the ‘subsonic limit’


regime, i.e., 0 < ε 1:

PGV ZS =

∫

Rd

i

2

d∑

j=1

(Ej ∇E∗

j − E∗j ∇Ej

)dx − αε2

ν

∫

RdNV dx

≈ PV NLS +O(ε2),

HGV ZS =

∫

Rd

[a ‖∇E‖2

l2 + (1 − a)|∇ · E|2 +αν − λ

2|E|4

]dx − αε2

2ν

∫

Rd|V|2 dx

≈ HV NLS +O(ε2).

2.3.4 Reduction from GZS to NLS

Similarly, in the “subsonic limit”, i.e. ε → 0 in GZS (3.18), (3.19), it collpases to thewell-known NLS equation with a cubic nonlinearity. In fact, letting ε → 0 in (3.19), weget formally

N = ν |E|2 + ε2 4−1 ∂ttN = ν |E|2 +O(ε2), when ε→ 0. (3.33)

Plugging (3.33) into (3.18), we obtain formally the NLS equation:

i Et + 4 E + (λ− αν)|E|2 E = 0, x ∈ Rd, t > 0. (3.34)

The NLS equation (3.34) is time reversible, time transverse invariant, and preserves thefollowing wave energy, momentum and Hamiltonian:

DNLS =

∫

Rd|E(x, t)|2 dx, (3.35)

PNLS =

∫

Rd

[i

2(E∇E∗ − E∗∇E)

]dx, (3.36)

HNLS =

∫

Rd

[|∇E|2 +

αν − λ

2|E|4

]dx. (3.37)

Similarly, letting ε → 0 in (3.21), (3.22), noting (3.33), we get formally the quadraticconvergence rate of the momentum and Hamiltonian from GZS to NLS in the ‘subsoniclimit’ regime, i.e., 0 < ε 1:

PGZS =

∫

Rd

i

2(E∇E∗ − E∗∇E) dx − ε2α

ν

∫

RdNV dx

≈ PNLS +O(ε2), (3.38)

HGZS =

∫

Rd

[|∇E|2 +

αν − λ

2|E|4

]dx− αε2

2ν

∫

Rd|V|2 dx

≈ HNLS +O(ε2). (3.39)

2.3.5 Add a linear damping term to arrest blowup

When d ≥ 2 and initial Hamiltonian HGZS < 0 in the GZS (3.18), (3.19), mathematically,it will blowup in finite time [113, 95]. However, the physical quantities modeled by E andN do not become infinite which implies the validity of (3.18), (3.19) breaks down nearsingularity. Additional physical mechanisms, which were initially small, become important

2.4. WELL-POSEDNESS OF ZS 61

near the singular point and prevent the formation of singularity. In order to arrest blowup,in physical literatures, a small linear damping (absorption) term is introduced into theGZS [64]:

i ∂tE + 4E − α N E + λ |E|2E + i γ E = 0, (3.40)

ε2 ∂ttN −4N + ν 4 |E|2 = 0, x ∈ Rd, t > 0; (3.41)

where γ > 0 is a damping parameter. The decay rate of the wave energy DGZS of thedamped GZS (3.40), (3.41) is

DGZS(t) =

∫

Rd|E(x, t)|2 dx = e−2γt

∫

Rd|E(x, 0)|2 dx = e−2γtDGZS(0), t ≥ 0. (3.42)

Similarly, when d ≥ 2 and initial Hamiltonian HGV ZS < 0 in the GVZS (3.8), (3.9)(or HV ZSM < 0 in the VZSM (3.1), (3.2)), mathematically, it will blowup in finite timetoo. In order to arrest blowup, in physical literatures, a small linear damping (absorption)term is introduced into the GVZS (or VZSM):

i ∂tE + a 4 E + (1 − a) ∇(∇ ·E) − α N E + λ |E|2E + i γ E = 0, (3.43)

ε2∂ttN −4N + ν 4 |E|2 = 0, x ∈ Rd, t > 0; (3.44)

where γ > 0 is a damping parameter. The decay rate of the wave energy DGV ZS of thedamped GVZS (3.43), (3.44) is

DGV ZS(t) =

∫

Rd|E(x, t)|2 dx = e−2γt

∫

Rd|E(x, 0)|2 dx

= e−2γtDGV ZS(0), t ≥ 0. (3.45)

2.4 Well-posedness of ZS

Based on the conservation laws, the wellposedness for the standard ZS (3.16)-(3.17) wereproven [113, 112, 23, 24]

Theorem 2.4.1 In one dimension, for initial conditions, E0 ∈ Hp(R), N0 ∈ Hp−1(R),and N (1) ∈ Hp−2(R) with p ≤ 3, there exists a unique solution E ∈ L∞(R+,Hp(R)),N ∈ L∞(R+,Hp−1(R)) for (3.16)-(3.17).

Theorem 2.4.2 In dimensions 2 and 3, for initial conditions E0 ∈ Hp(Rd), N0 ∈Hp−1(Rd), and N (1) ∈ Hp−2(Rd) with p ≤ 3, there exists a unique solution E ∈ L∞([0, T ∗),Hp(Rd)),N ∈ L∞([0, T ∗), Hp−1(Rd)) for (3.16)-(3.17), where time T ∗ depends on the initial con-ditions.

2.5 Plane wave and soliton wave solutions

In one spatial dimension (1D), the GZS (3.40)- (3.41) collapses to

i Et + Exx − αN E + λ|E|2 E + iγ E = 0, a < x < b, t > 0, (5.1)

ε2Ntt −Nxx + ν(|E|2)xx = 0, a < x < b, t > 0, (5.2)


which admits plane wave and soliton wave solutions.

Firstly, it is instructive to examine some explicit solutions to (5.1) and (5.2). Thewell-known plane wave solutions [97] can be given in the following form:

N(x, t) = d, a < x < b, t ≥ 0, (5.3)

E(x, t) =

c ei(2πrxb−a

−ωt), ω = αd+ 4π2r2

(b−a)2 − λc2, γ = 0,

c e−γtei

(2πrxb−a

−ωt−λc2

2γ(e−2γt−1)

), ω = αd+ 4π2r2

(b−a)2 , γ 6= 0,(5.4)

where r is an integer and c, d are constants.

Secondly, as is well known, the standard ZS is not exactly integrable. Therefore thegeneralized ZS cannot be exactly integrable, either. However, it has exact one-solitonsolutions to (5.1) and (5.2) for γ = 0 [72, 73]:

Es(x, t; η, V, ε, ν) =

[λ

2− αν

2ε2(1/ε2 − V 2)−1

]−1/2

Us, x ∈ R, t ≥ 0, (5.5)

Us ≡ 2iη sech[2η(x − V t)] exp[iV x/2 + i(4η2 − V 2/4)t + iΦ0

], (5.6)

Ns(x, t; η, V, ε, ν) =ν

ε2(1/ε2 − V 2)−1|Es|2, (5.7)

where η and V are the soliton’s amplitude and velocity, respectively, and Φ0 is a trivialphase constant.

Finally, we will consider the periodic soliton solution with a period L in 1d of thestandard ZS, that is, d = 1, ε = 1, α = 1, λ = 0, γ = 0 and ν = −1 in (3.40)-(3.41).The analytic solution of the ZS (5.1)-(5.2) was derived [100, 62] and used to test differentnumerical methods for the ZS in [100, 62, 28]. The solution can be written as

Es(x, t; v,Emax) = F (x− vt) exp[iφ(x− ut)], (5.8)

Ns(x, t; v,Emax) = G(x− vt), (5.9)

where

F (x− vt) = Emax · dn(w, q), G(x− vt) =|F (x− vt)|2v2 − 1

+N0,

w =Emax√

(2(1 − v2))· (x− vt), q =

√(E2

max − E2min)

Emax,

φ = v/2,v

2L = 2πm, m = 1, 2, 3 · · · , u =

v

2+

2N0

v− E2

max + E2min

v(1 − v2),

L =2√

2(1 − v2)

EmaxK(q) =

2√

2(1 − v2)

EmaxK ′(Emin

Emax

),

with dn(w, q) a Jacobian elliptic function, L the period of the Jacobian elliptic functions,

K and K ′ the complete elliptic integrals of the first kind satisfying K(q) = K ′(√

1 − q2),

and N0 chosen such that 〈Ns〉 = 1L

∫ L

0Ns(x, t) dx = 0.

2.6. TIME-SPLITTING SPECTRAL METHOD 63

2.6 Time-splitting spectral method

In this section we present new numerical methods for the GZS (3.40), (3.41). For simplicityof notations, we shall introduce the method in one space dimension (d = 1) of the GZSwith periodic boundary conditions. Generalizations to d > 1 are straightforward for tensorproduct grids and the results remain valid without modifications. For d = 1, the problembecomes

i ∂tE + ∂xxE − αN E + λ|E|2 E + iγ E = 0, a < x < b, t > 0, (6.1)

ε2∂ttN − ∂xx(N − ν |E|2) = 0, a < x < b, t > 0, (6.2)

E(x, 0) = E(0)(x), N(x, 0) = N (0)(x), ∂tN(x, 0) = N (1)(x), a ≤ x ≤ b, (6.3)

E(a, t) = E(b, t), ∂xE(a, t) = ∂xE(b, t), t ≥ 0, (6.4)

N(a, t) = N(b, t), ∂xN(a, t) = ∂xN(b, t), t ≥ 0. (6.5)

Moreover, we supplement (6.1)-(6.5) by imposing the compatibility condition

E(0)(a) = E(0)(b), N (0)(a) = N (0)(b), N (1)(a) = N (1)(b),

∫ b

aN (1)(x) dx = 0. (6.6)

As is well known, the GZS has the following property

DGZS(t) =

∫ b

a|E(x, t)|2 dx = e−2γt

∫ b

a|E(0)(x)|2 dx = e−2γtDGZS(0), t ≥ 0. (6.7)

When γ = 0, DGZS(t) ≡ DGZS(0), i.e., it is an invariant of the GZS [28]. When γ > 0, itdecays to 0 exponentially. Furthermore, the GZS also has the following properties

∫ b

a∂tN(x, t) dx = 0,

∫ b

aN(x, t) dx =

∫ b

aN (0)(x) dx = const., t ≥ 0. (6.8)

In some cases, the boundary conditions (6.4) and (6.5) may be replaced by

E(a, t) = E(b, t) = 0, N(a, t) = N(b, t) = 0, t ≥ 0. (6.9)

We choose the spatial mesh size h = 4x > 0 with h = (b− a)/M for M being an evenpositive integer, the time step k = 4t > 0 and let the grid points and the time step be

xj := a+ j h, j = 0, 1, · · · ,M ; tm := m k, m = 0, 1, 2, · · · .

Let Emj and Nmj be the approximations of E(xj , tm) and N(xj, tm), respectively. Further-

more, let Em and Nm be the solution vector at time t = tm = mk with components Emjand Nm

j , respectively.

From time t = tm to t = tm+1, the first NLS-type equation (6.1) is solved in twosplitting steps. One solves first

i ∂tE + ∂xxE = 0, (6.10)

for the time step of length k, followed by solving

i ∂tE = αN E − λ|E|2 E − iγ E, (6.11)


for the same time step. Equation (6.10) will be discretized in space by the Fourier spectralmethod and integrated in time exactly. For t ∈ [tm, tm+1], multiplying (6.11) by E, we get

i ∂tE E∗ = αN |E|2 − λ|E|4 − iγ|E|2. (6.12)

Then calculating the conjugate of the ODE (6.11) and multiplying it by E, one finds

−i ∂tE∗ E = αN |E|2 − λ|E|4 + iγ|E|2. (6.13)

Subtracting (6.13) from (6.12) and then multiplying both sides by −i, one gets

∂t(|E(x, t)|2) = ∂tE(x, t)E(x, t)∗ + ∂tE(x, t)∗E(x, t) = −2γ|E(x, t)|2 (6.14)

and therefore

|E(x, t)|2 = e−2γ(t−tm)|E(x, tm)|2, tm ≤ t ≤ tm+1. (6.15)

Substituting (6.15) into (6.11), we obtain

i∂tE(x, t) = αN(x, t)E(x, t) − λe−2γ(t−tm)|E(x, tm)|2E(x, t) − iγE(x, t). (6.16)

Integrating (6.16) from tm to tm+1, and then approximating the integral of N on [tm, tm+1]via the trapezoidal rule, one obtains

E(x, tm+1) = e−i∫ tm+1

tm[αN(x,τ)−λe−2γ(τ−tm)|E(x,tm)|2−iγ] dτ

E(x, tm)

≈e−ik[α(N(x,tm)+N(x,tm+1))/2−λ|E(x,tm)|2] E(x, tm), γ = 0,

e−γk−i[kα(N(x,tm)+N(x,tm+1))/2+λ|E(x,tm)|2(e−2γk−1)/2γ] E(x, tm), γ 6= 0.

2.6.1 Crank-Nicolson leap-frog time-splitting spectral discretizations (CN-LF-TSSP)

The second wave-type equation (6.2) in the GZS is discretized by pseudo-spectral methodfor spatial derivatives, and then applying Crank-Nicolson /leap-frog for linear/nonlinearterms for time derivatives:

ε2Nm+1j − 2Nm

j +Nm−1j

k2−Df

xx

[(βNm+1 + (1 − 2β)Nm + βNm−1

)− ν|Em|2

]x=xj

= 0, j = 0, · · · ,M, m = 1, 2, · · · , (6.17)

where 0 ≤ β ≤ 1 is a constant, Dfxx, a spectral differential operator approximation of ∂xx,

is defined as

DfxxU

∣∣∣x=xj

= −M/2−1∑

l=−M/2

µ2l Ul e

iµl(xj−a) (6.18)

and Ul, the Fourier coefficients of a vector U = (U0, U1, U2, · · · , UM )T with U0 = UM , aredefined as

µl =2πl

b− a, Ul =

1

M

M−1∑

j=0

Uj e−iµl(xj−a), l = −M

2, · · · , M

2− 1. (6.19)


When β = 0 in (6.17), the discretization (6.17) to the wave-type equation (6.2) is explicitand was used in [15, 114]. When 0 < β ≤ 1, the discretization is implicit, but can besolved explicitly. In fact, suppose

Nmj =

M/2−1∑

l=−M/2

(Nm)l eiµl(xj−a), j = 0, · · · ,M ; m = 0, 1, · · · , (6.20)

Plugging (6.20) into (6.17), using the orthogonality of the Fourier basis, we obtain

ε2˜(Nm+1)l − 2(Nm)l +

˜(Nm−1)lk2

+ µ2l

[β ˜(Nm+1)l + (1 − 2β)(Nm)l + β ˜(Nm−1)l

−ν ˜(|Em|2)l]

= 0, l = −M/2, · · · ,M/2 − 1; m = 1, 2, · · · . (6.21)

Solving the above equation, we get

˜(Nm+1)l =

(2 − k2µ2

l

ε2 + βk2µ2l

)(Nm)l − (Nm−1)l +

νk2µ2l

ε2 + βk2µ2l

˜(|Em|2)l,

l = −M/2, · · · ,M/2 − 1; m = 1, 2, · · · . (6.22)

From time t = tm to t = tm+1, we combine the splitting steps via the standard Strangsplitting:

Nm+1j =

M/2−1∑

l=−M/2

˜(Nm+1)l eiµl(xj−a), (6.23)

E∗j =

M/2−1∑

l=−M/2

e−ikµ2l/2(Em)l e

iµl(xj−a),

E∗∗j =

e−ik[α(Nm

j +Nm+1j )/2−λ|E∗

j |2] E∗

j , γ = 0,

e−γk−i[kα(Nmj +Nm+1

j )/2+λ|E∗j |

2(e−2γk−1)/2γ] E∗j , γ 6= 0,

Em+1j =

M/2−1∑

l=−M/2

e−ikµ2l /2(E∗∗)l e

iµl(xj−a), 0 ≤ j ≤M − 1, m ≥ 0; (6.24)

where ˜(Nm+1)l is given in (6.22) for m > 0 and (6.27) for m = 0. The initial conditions(6.3) are discretized as

E0j = E(0)(xj), N

0j = N (0)(xj),

N1j −N−1

j

2k= N

(1)j , j = 0, 1, 2, · · · ,M − 1, (6.25)

where

N(1)j =

N (1)(xj), 0 ≤ j ≤M − 2,

−M−2∑

l=0

N (1)(xl), j = M − 1.(6.26)

This implies that

(N1)l =

(1 − k2µ2

l

2(ε2 + βk2µ2l )

)(N (0))l + k (N (1))l +

νk2µ2l

2(ε2 + βk2µ2l )

˜(|E(0)|2)l,

l = −M/2, · · · ,M/2 − 1. (6.27)


This type of discretization for the initial condition (6.3) is equivalent to use the trapezoidalrule for the periodic function N (1) and such that (6.8) is satisfied in discretized level. Thediscretization error converges to 0 exponentially fast as the mesh size h goes to 0.

Note that the spatial discretization error of the method is of spectral-order accuracyin h and time discretization error is demonstrated to be second-order accuracy in k fromour numerical results.

2.6.2 Phase space analytical solver + time-splitting spectral discretiza-tions (PSAS-TSSP)

Another way to discretize the second wave-type equation (6.2) in GZS is by pseudo-spectralmethod for spatial derivatives, and then solving the ODEs in phase space analyticallyunder appropriate chosen transmission conditions between different time intervals. Fromtime t = tm to t = tm+1, assume

N(x, t) =

M/2−1∑

l=−M/2

Nml (t) eiµl(x−a), a ≤ x ≤ b, tm ≤ t ≤ tm+1. (6.28)

Plugging (6.28) into (6.2) and noticing the orthogonality of the Fourier series, we get thefollowing ODEs:

ε2d2Nm

l (t)

d t2+ µ2

l

[Nml (t) − ν ˜(|E(tm)|2)l

]= 0, tm ≤ t ≤ tm+1, m ≥ 0, (6.29)

Nml (tm) =

(N (0))l, m = 0,

Nm−1l (tm), m > 0,

l = −M/2, · · · ,M/2 − 1. (6.30)

For each fixed l (−M/2 ≤ l ≤ M/2 − 1), Eq. (6.29) is a second-order ODE. It needs twoinitial conditions such that the solution is unique. When m = 0 in (6.29), (6.30), we havethe initial condition (6.30) and we can pose the other initial condition for (6.29) due tothe initial condition (6.3) for the GZS (6.1)-(6.5):

d

dtN0l (t0) =

d

dtN0l (0) = (N (1))l, l = −M/2, · · · ,M/2 − 1. (6.31)

Then the solution of (6.29), (6.30) with m = 0 and (6.31) is:

N0l (t) =

(N (0))0 + t (N (1))0, l = 0,

[(N (0))l − ν ˜(|E(0)|2)l

]cos(µl t/ε) + ν ˜(|E(0)|2)l l 6= 0,

+ εµl

(N (1))l sin(µl t/ε),

(6.32)

0 ≤ t ≤ t1, l = −M/2, · · · ,M/2 − 1.

But when m > 0, we only have one initial condition (6.30). One can’t simply pose thecontinuity between d

dtNml (t) and d

dtNm−1l (t) across the time t = tm due to the last term

in (6.29) is usually different in two adjacent time intervals [tm−1, tm] and [tm, tm+1], i.e.


˜(|E(tm−1)|2)l 6= ˜(|E(tm)|2)l. Since our goal is to develop explicit scheme and we needlinearize the nonlinear term in (6.2) in our discretization (6.29), in general,

d

dtNm−1l (t−m) = lim

t→t−m

d

dtNm−1l (t) 6= lim

t→t+m

d

dtNml (t) =

d

dtNml (t+m), (6.33)

m = 1, · · · , l = −M/2, · · · ,M/2 − 1.

Unfortunely, we don’t know the jump ddtN

ml (t+m) − d

dtNm−1l (t−m) across the time t = tm.

In order to get a unique solution of (6.29), (6.30) for m > 0, here we pose an additionalcondition:

Nml (tm−1) = Nm−1

l (tm−1), l = −M/2, · · · ,M/2 − 1. (6.34)

The condition (6.34) is equivalent to pose the solution Nml (t) on the time interval [tm, tm+1]

of (6.29), (6.30) is also continuity at the time t = tm−1. After a simple computation, weget the solution of (6.29), (6.30) and (6.34) for m > 0:

Nml (t) =

Nm−10 (tm) + t−tm

k

[Nm−1

0 (tm) − Nm−10 (tm−1)

], l = 0,

[Nm−1l (tm) − ν ˜(|Em|2)l

]cos(µl(t− tm)/ε) l 6= 0,

+ν ˜(|Em|2)l +sin(µl(t−tm)/ε)

sin(kµl/ε)

[Nm−1l (tm) cos(kµl/ε)

−Nm−1l (tm−1) + ν [1 − cos(kµl/ε)] ˜(|Em|2)l

],

(6.35)

tm ≤ t ≤ tm+1, l = −M/2, · · · ,M/2 − 1.

From time t = tm to t = tm+1, we combine the splitting steps via the standard Strangsplitting:

Nm+1j =

M/2−1∑

l=−M/2

Nml (tm+1) e

iµl(xj−a), (6.36)

E∗j =

M/2−1∑

l=−M/2

e−ikµ2l/2(Em)l e

iµl(xj−a),

E∗∗j =

e−ik[α(Nm

j +Nm+1j )/2−λ|E∗

j |2] E∗

j , γ = 0,

e−γk−i[kα(Nmj +Nm+1

j )/2+λ|E∗j |

2(e−2γk−1)/2γ] E∗j , γ 6= 0,

Em+1j =

M/2−1∑

l=−M/2

e−ikµ2l /2(E∗∗)l e

iµl(xj−a), 0 ≤ j ≤M − 1, m ≥ 0; (6.37)

where

Nml (tm+1) =

(N (0))0 + k (N (1))0, l = 0, m = 0,

(N (0))l cos(kµl/ε) + εµl

(N (1))l sin(kµl/ε) l 6= 0, m = 0,

+ν [1 − cos(kµl/ε)] ˜(|E(0)|2)l,

2Nm−1l (tm) cos(kµl/ε) − Nm−1

l (tm−1) m ≥ 1,

+2ν [1 − cos(kµl/ε)] ˜(|Em|2)l,

(6.38)


The initial conditions (6.3) are discretized as

E0j = E(0)(xj), N

0j = N (0)(xj), (∂tN)0j = N

(1)j , j = 0, 1, 2, · · · ,M − 1. (6.39)

Note that the spatial discretization error of the above method is again of spectral-orderaccuracy in h and time discretization error is demonstrated to be second-order accuracyin k from our numerical results.

2.6.3 Properties of the numerical methods

(1). Plane wave solution: If the initial data in (6.3) is chosen as

E(0)(x) = c ei2πlx/(b−a), N (0)(x) = d, N (1)(x) = 0, a ≤ x ≤ b, (6.40)

where l is an integer and c, d are constants, then the GZS (6.1)-(6.5) admits the planewave solution [97]

N(x, t) = d, a < x < b, t ≥ 0, (6.41)

E(x, t) =

c ei(2πlxb−a

−ωt), ω = αd+ 4π2l2

(b−a)2 − λc2, γ = 0,

c e−γtei

(2πlxb−a

−ωt−λc2

2γ(e−2γt−1)

), ω = αd+ 4π2l2

(b−a)2 , γ 6= 0.(6.42)

It is easy to see that in this case our numerical methods CN-LF-TSSP (6.23), (6.24) andPAAS-TSSP (6.36), (6.37) give exact results provided that M ≥ 2(|l| + 1).(2). Time transverse invariant: A main advantage of CN-LF-TSSP and PAAS-TSSPis that if a constant r is added to the initial data N0(x) in (6.3) when γ = 0 in (6.1),then the discrete functions Nm+1

j obtained from (6.23) or (6.36) get added by r and Em+1j

obtained from (6.24) or (6.37) get multiplied by the phase factor e−ir(m+1)k, which leavesthe discrete function |Em+1

j |2 unchanged. This property also holds for the exact solutionof GZS, but does not hold for the finite difference schemes proposed in [62, 28] and thespectral method proposed in [100].(3). Conservation: Let U = (U0, U1, · · · , UM )T with U0 = UM , f(x) a periodic functionon the interval [a, b], and let ‖ · ‖l2 be the usual discrete l2-norm on the interval (a, b), i.e.,

‖U‖l2 =

√√√√b− a

M

M−1∑

j=0

|Uj |2, ‖f‖l2 =

√√√√b− a

M

M−1∑

j=0

|f(xj)|2. (6.43)

Then we have

Theorem 2.6.1 The CN-LF-TSSP (6.23), (6.24) and PSAS-TSSP (6.36), (6.37) forGZS possesses the following properties (in fact, they are the discretized version of (6.7)and (6.8)):

‖Em‖2l2 = e−2γtm‖E0‖2

l2 = e−2γtm‖E(0)‖2l2 m = 0, 1, 2, · · · , (6.44)

b− a

M

M−1∑

j=0

Nm+1j −Nm

j

k= 0, m = 0, 1, 2, · · · (6.45)

andb− a

M

M−1∑

j=0

Nmj =

b− a

M

M−1∑

j=0

N0j =

b− a

M

M−1∑

j=0

N (0)(xj), m ≥ 0. (6.46)


Proof: From (6.43), (6.37) and (6.19), using the orthogonality of the discrete Fourierseries and noticing the Pasavel equality, we have

M

b− a‖Em+1‖2

l2 =M−1∑

j=0

|Em+1j |2 = M

M/2−1∑

l=−M/2

∣∣∣e−ikµ2l/2(E∗∗)l

∣∣∣2

= M

M/2−1∑

l=−M/2

|(E∗∗)l|2

=M−1∑

j=0

|E∗∗j |2 = e−2γk

M−1∑

j=0

|E∗j |2 = e−2γk

M−1∑

j=0

|Emj |2

= e−2γk M

b− a‖Em‖2

l2 , m ≥ 0. (6.47)

Thus (6.44) is obtained from (6.47) by induction. The equalities (6.45) and (6.46) can beobtained in a similar way.

(4). Unconditional stability: By the standard Von Neumann analysis for (6.23) and(6.36), noting (6.44), we get PSAS-TSSP and CN-LF-TSSP with 1/4 ≤ β ≤ 1 are uncon-ditionally stable, and CN-LF-TSSP with 0 ≤ β < 1/4 is conditionally stable with stabilityconstraint k ≤ 2hε

π√d(1−4β)

in d-dimensions ( d = 1, 2, 3). In fact, for PSAS-TSSP (6.36),

(6.37), setting ˜(|Em|2)l = 0 and plugging Nml (tm+1) = µNm−1

l (tm) = µ2Nm−1l (tm−1) into

(6.38) with |µ| the amplification factor, we obtain the characteristic equation:

µ2 − 2 cos(kµl/ε)µ+ 1 = 0. (6.48)

This implies

µ = cos(kµl/ε) ± i sin(kµl/ε). (6.49)

Thus the amplification factor

Gl = |µ| =√

cos2(kµl/ε) + sin2(kµl/ε) = 1, l = −M/2, · · ·M/2 − 1.

This together with (6.44) imply that PSAS-TSSP is unconditionally stable. Similarly forCN-LF-TSSP (6.23), (6.24), noting (6.22), we have the characteristic equation:

µ2 −(

2 − k2µ2l

ε2 + βk2µ2l

)µ+ 1 = 0. (6.50)

This implies

µ = 1 − k2µ2l

2(ε2 + βk2µ2l )

±

√√√√(

1 − k2µ2l

2(ε2 + βk2µ2l )

)2

− 1. (6.51)

When 1/4 ≤ β ≤ 1, we have

∣∣∣∣∣1 − k2µ2l

2(ε2 + βk2µ2l )

∣∣∣∣∣ ≤ 1, k > 0, l = −M/2, · · ·M/2 − 1.

Thus

µ = 1 − k2µ2l

2(ε2 + βk2µ2l )

± i

√√√√1 −(

1 − k2µ2l

2(ε2 + βk2µ2l )

)2

. (6.52)


This implies the amplification factor

Gl = |µ| =

√√√√(

1 − k2µ2l

2(ε2 + βk2µ2l )

)2

+ 1 −(

1 − k2µ2l

2(ε2 + βk2µ2l )

)2

= 1, l = −M/2, · · ·M/2 − 1.

This together with (6.44) imply that CN-LF-TSSP with 1/4 ≤ β ≤ 1/2 is unconditionallystable. On the other hand, when 0 ≤ β < 1/4, we need the stability condition

∣∣∣∣∣1 − k2µ2l

2(ε2 + βk2µ2l )

∣∣∣∣∣ ≤ 1 =⇒ k ≤ min−M/2≤l≤M/2−1

2ε√(1 − 4β)µ2

l

=2hε

π√

1 − 4β.

This together with (6.44) imply that CN-LF-TSSP with 0 ≤ β < 1/4 is conditionallystable in one dimension with stability condition

k ≤ 2hε

π√

1 − 4β. (6.53)

All above stability results are confirmed by our numerical experiments.

(5). ε-resolution in the ‘subsonic limit’ regime (0 < ε 1): As our numericalresults suggest: The meshing strategy (or ε-resolution) which guarantees good numericalapproximations of our new numerical methods PSAS-TSSP and CN-LF-TSSP with 1/4 ≤β ≤ 1/2 in the ‘subsonic limit’ regime, i.e. 0 < ε 1, is: i). for initial data withO(ε)-wavelength: h = O(ε) and k = O(ε); ii). for initial data with O(1)-wavelength:h = O(1) and k = O(1). Where the meshing strategy for CN-LF-TSSP with 0 ≤ β < 1/4is: h = O(ε) & k = O(hε) = O(ε2); h = O(1) & k = O(ε), respectively.

Remark 2.6.1 If the periodic boundary conditions (6.4) and (6.5) are replaced by thehomogeneous Dirichlet boundary condition (6.9), then the Fourier basis used in the abovealgorithm can be replaced by the sine basis [15] or the algorithm in section 4 for VZSM.Similarly, if homogeneous Neumann conditions are used, then cosine series can be appliedin designing the algorithm.

2.6.4 Extension TSSP to GVZS

The idea to construct the numerical methods CN-LF-TSSP and PSAS-TSSP for GZS(6.1)-(6.5) can be easily extended to the VZSM [114] in three dimensions for M differentacoustic modes in a box Ω = [a1, b1]×[a2, b2]×[a3, b3] with homogeneous Dirichlet boundaryconditions:

i∂tE + a 4 E + (1 − a) ∇(∇ · E) − αEM∑

J=1

NJ + λ|E|2E + iγE = 0, (6.54)

ε2J∂ttNJ −4NJ + νJ 4 |E|2 = 0, J = 1, · · · ,M, x ∈ Ω, t > 0, (6.55)

E(x, 0) = E(0)(x), NJ(x, 0) = N(0)J (x), ∂tNJ(x, 0) = N

(1)J (x), x ∈ Ω, (6.56)

E(x, t) = 0, NJ(x, t) = 0 (J = 1, · · · ,M), x ∈ ∂Ω; (6.57)


where x = (x, y, z)T and E(x, t) = (E1(x, t), E2(x, t), E3(x, t))T . Moreover, we supplement

(6.54)-(6.57) by imposing the compatibility condition

E(0)(x) = 0, N(0)J (x) = N

(1)J (x) = 0, x ∈ ∂Ω, J = 1, · · · ,M. (6.58)

In some cases, the homogeneous Dirichlet boundary condition (6.57) may be replacedby periodic boundary conditions:

With periodic boundary conditions for E, NJ(J = 1, · · · ,M) on ∂Ω. (6.59)

We choose the spatial mesh sizes h1 = b1−a1M1

, h2 = b2−a2M2

and h3 = b3−a3M3

in x-, y-and z-direction respectively, with M1, M2 and M3 given positive integers; the time stepk = 4t > 0. Denote grid points and time steps as

xj := a1 + jh1, j = 0, 1, · · · ,M1; yp := a2 + ph2, p = 0, 1, · · · ,M2;

zs := a3 + sh3, s = 0, 1, · · · ,M3; tm := mk, m = 0, 1, 2, · · · .

Let Emj,p,s and (NJ)

mj,p,s be the approximations of E(xj , yp, zs, tm) and NJ(xj , yp, zs, tm),

respectively.

For simplicity, here we only extend PSAS-TSSP from GZS (6.1)-(6.5) to VZSM (6.54)-(6.57) with homogeneous Dirichlet conditions. For periodic boundary conditions (6.59) orextension of CN-LF-TSSP can be done in a similar way. Following the idea of constructingPSAS-TSSP for GZS and the TSSP for VZSM in [114] here we only present the numericalalgorithm. From time t = tm to t = tm+1, the PSAS-TSSP method for VZSM (6.54)-(6.57)reads:

(NJ)m+1j,p,s =

∑

(l,g,r)∈N

(NJ)m

l,g,r(tm+1) sin

(ljπ

M1

)sin

(pgπ

M2

)sin

(srπ

M3

), (6.60)

E∗j,p,s =

∑

(l,g,r)∈N

Bl,g,r(k/2) (Em)l,g,r sin

(ljπ

M1

)sin

(pgπ

M2

)sin

(srπ

M3

),

E∗∗j,p,s =

e−ik[α∑M

J=1((NJ )m

j,p,s+(NJ )m+1j,p,s )/2−λ|E∗

j,p,s|2] E∗

j,p,s, γ = 0,

e−γk−i[kα∑M

J=1((NJ )m

j,p,s+(NJ )m+1j,p,s )/2+λ|E∗

j,p,s|2(e−2γk−1)/2γ] E∗

j,p,s, γ 6= 0,

Em+1j,p,s =

∑

(l,g,r)∈N

Bl,g,r(k/2) (E∗∗)l,g,r sin

(ljπ

M1

)sin

(pgπ

M2

)sin

(srπ

M3

), (6.61)

where

N = (l, g, r) | 1 ≤ l ≤M1 − 1, 1 ≤ g ≤M2 − 1, 1 ≤ r ≤M3 − 1.


(NJ )m

l,g,r(tm+1) =

˜(N

(0)J )0,0,0 + k

˜(N

(1)J )0,0,0, Rl,g,r = 0,m = 0,

˜(N

(0)J )l,g,r cos(kRl,g,r/εJ ) Rl,g,r 6= 0,m = 0,

+ εJRl,g,r

˜(N

(1)J )l,g,r sin(kRl,g,r/εJ )

+νJ [1 − cos(kRl,g,r/εJ)] ˜(|E(0)|2)l,g,r,

2(NJ )m−1

l,g,r (tm) cos(kRl,g,r/εJ ) m ≥ 1,

+2νJ [1 − cos(kRl,g,r/εJ )] ˜(|Em|2)l,g,r−(NJ)

m−1

l,g,r (tm−1),

and

Bl,g,r(τ) =

I3, l = g = r = 0,

e−iaτR2l,g,r

[I3 +

e−i(1−a)τR2l,g,r − 1

R2l,g,r

Al,g,r

], otherwise,

with

R2l,g,r = κ2

l + ζ2g + η2

r , Al,g,r =

κ2l κlζg κlηr

κlζg ζ2g ζgηr

κlηr ζgηr η2r

=

κlζgηr

(κl ζg ηr

);

where I3 is the 3×3 identity matrix, and Ul,g,r, the sine-transform coefficients, are definedas

Ul,g,r =2

M1

2

M2

2

M3

M1−1∑

j=1

M2−1∑

p=1

M3−1∑

s=1

Uj,p,s sin

(ljπ

M1

)sin

(pgπ

M2

)sin

(srπ

M3

), (6.62)

with

κl =πl

b1 − a1, l = 1, . . . ,M1 − 1, ζg =

πg

b2 − a2, g = 1, . . . ,M2 − 1,

ηr =πr

b3 − a3, r = 1, . . . ,M3 − 1.

The initial conditions (6.56) are discretized as

E0j,p,s = E(0)(xj , yp, zs),

(NJ)0j,p,s = N

(0)J (xj , yp, zs), j = 0, · · · ,M1, p = 0, · · · ,M2, s = 0, · · · ,M3,

(∂tNJ)0j,p,s = N

(1)J (xj , yp, zs), J = 1, · · · ,M.

The properties of the numerical method for GZS in section 3 are still valid here.

2.7. CRANK-NICOLSON FINITE DIFFERENCE (CNFD) METHOD 73

2.7 Crank-Nicolson finite difference (CNFD) method

Another method for the GZS (3.40)-(3.41) is to use centered finite difference for spatialderivatives and Crank-Nicolson for time derivative. For simplicity of notations , herewe only present the CNFD method for the standard ZS [28] with homogeneous Dirichletboundary condition (6.9), i.e., in (6.1)-(6.2) with ε = 1, ν = −1, α = 1, λ = 0 and γ = 0:

iEm+1j − Emj

k+

1

2

(Em+1j+1 − 2Em+1

j + Em+1j−1

h2+Emj+1 − 2Emj + Emj−1

h2

)

=1

4(Nm

j +Nm+1j )(Em+1

j + Emj ), j = 1, 2, · · · ,M − 1, (7.1)

Nm+1j − 2Nm

j +Nm−1j

k2− θ

(Nm+1j+1 − 2Nm+1

j +Nm+1j−1

h2+Nm−1j+1 − 2Nm−1

j +Nm−1j−1

h2

)

−(1 − 2θ)Nmj+1 − 2Nm

j +Nmj−1

h2=

|Emj+1|2 − 2|Emj |2 + |Emj−1|2h2

, (7.2)

Em+10 = Em+1

M = 0, Nm+10 = Nm+1

0 = 0, m = 0, 1 · · · . (7.3)

where 0 ≤ θ ≤ 1 is a parameter. The initial conditions are discretized as:

E0j = E0(xj), N0

j = N0(xj), j = 0, 1, · · · ,M, (7.4)

N1j = N0

j + kN1(xj) +k2

2

[N0j+1 − 2N0

j +N0j−1

h2

+|E0j+1|2 − 2|E0

j |2 + |E0j−1|2

h2

]. (7.5)

When θ = 0, the discretization (7.2) for wave-type equation is explicit; when θ > 0,it is implicit but can be solved explicitly when periodic boundary conditions are applied.Generalization of the method to GZS are straightforward. In our computations in nextsubsection, we choose θ = 0.5.

Remark 2.7.1 In [62, 63], convergence and error estimate of the CNFD discretization(7.1), (7.2) are proved.

2.8 Numerical results

In this section, we present numerical results of GZS with a solitary wave solution in onedimension to compare the accuracy, stability and ε-resolution of different methods. Wealso present numerical examples solitary-wave collisions in one dimension GZS.

In our computation, the initial conditions for (6.3) are always chosen such that |E0|,N0 and N (1) decay to zero sufficiently fast as |x| → ∞. We always compute on a domain,which is large enough such that the periodic boundary conditions do not introduce asignificant aliasing error relative to the problem in the whole space.

Example 2.1 The standard ZS with a solitary-wave solution, i.e., we choose d = 1,α = 1, λ = 0, γ = 0 and ν = −1 in (3.40)-(3.41). The well-known solitary-wave solution


(5.5)-(5.7) of the ZS in this case is given in [97, 73]

E(x, t) =√

2B2(1 − ε2C2) sech(B(x− Ct)) ei[(C/2)x−((C/2)2−B2)t], (8.1)

N(x, t) = −2B2 sech2(B(x− Ct)), −∞ < x <∞, t ≥ 0, (8.2)

where B, C are constants. The initial condition is taken as

E(0)(x) = E(x, 0), N (0)(x) = N(x, 0), N (1)(x, 0) = ∂tN(x, 0), −∞ < x <∞, (8.3)

where E(x, 0), N(x, 0) and ∂tN(x, 0) are obtained from (8.1), (8.2) by setting t = 0.

We present computations for two different regimes of the acoustic speed, i.e. 1/ε:

Case I. O(1)-acoustic speed, i.e. we choose ε = 1, B = 1, C = 0.5 in (8.1), (8.2).Here we test the spatial and temporal discretization errors, conservation of the conservedquantities as well as the stability constraint of different numerical methods. We solvethe problem on the interval [-32, 32], i.e., a = −32 and b = 32 with periodic boundaryconditions. Let Eh,k and Nh,k be the numerical solution of (6.1), (6.5) in one dimensionwith the initial condition (8.3) by using a numerical method with mesh size h and timestep k. To quantify the numerical methods, we define the error functions as

e1 = ‖E(·, t) − Eh,k(t)‖l2 , e2 = ‖N(·, t) −Nh,k(t)‖l2 ,

e =‖E(·, t) − Eh,k(t)‖l2

‖E(·, t)‖l2+

‖N(·, t) −Nh,k(t)‖l2‖N(·, t)‖l2

=e1

‖E(·, t)‖l2+

e2‖N(·, t)‖l2

and evaluate the conserved quantities DGZS, PGZS and HGZS by using the numericalsolution, i.e. replacing E andN by their numerical counterparts Eh,k andNh,k respectively,in (3.20)-(3.22).

First, we test the discretization error in space. In order to do this, we choose a verysmall time step, e.g., k = 0.0001 such that the error from time discretization is negligiblecomparing to the spatial discretization error, and solve the ZS with different methodsunder different mesh sizes h. Table 2.1 lists the numerical errors of e1 and e2 at t = 2.0with different mesh sizes h for different numerical methods.

Mesh h = 1.0 h = 12 h = 1

4

PSAS-TSSPe1e2

9.810E-20.143

1.500E-41.168E-3

8.958E-96.500E-8

CN-LF-TSSP(β = 0)e1e2

9.810E-20.143

1.500E-41.168E-3

7.409E-93.904E-8

CN-LF-TSSP(β = 1/4)e1e2

9.810E-20.143

1.500E-41.168E-3

8.628E-96.521E-8

CN-LF-TSSP(β = 1/2)e1e2

9.810E-20.143

1.500E-41.168E-3

1.098E-86.326E-8

CNFDe1e2

0.4910.889

0.1200.209

2.818E-24.726E-2

Table 2.1: Spatial discretization error analysis: e1, e2 at time t=2 under k = 0.0001.

2.8. NUMERICAL RESULTS 75

Secondly, we test the discretization error in time. Table 2.2 shows the numerical errorsof e1 and e2 at t = 2.0 under different time steps k and mesh sizes h for different numericalmethods.

h Error k = 1100 k = 1

400 k = 11600 k = 1

6400

PSAS-TSSP 14 e1 4.968E-5 3.109E-6 1.944E-7 1.226E-8

e2 1.225E-4 7.664E-6 4.797E-7 3.871E-8

18 e1 4.968E-5 3.109E-6 1.944E-7 1.172E-8

e2 1.225E-4 7.664E-6 4.797E-7 3.157E-8

CN-LF-TSSP(β = 0) 14 e1 4.829E-5 3.022E-6 1.888E-7 1.156E-8

e2 1.032E-4 6.456E-6 4.041E-7 3.673E-8

18 e1 4.829E-5 3.022E-6 1.888E-7 1.100E-8

e2 1.032E-4 6.456E-6 4.043E-7 2.946E-8

CN-LF-TSSP(β = 1/4) 14 e1 5.679E-5 3.556E-6 2.224E-7 1.425E-8

e2 1.623E-4 1.015E-5 6.351E-7 4.970E-8

18 e1 5.679E-5 3.556E-6 2.224E-7 1.377E-8

e2 1.623E-4 1.015E-5 6.351E-7 4.356E-8

CN-LF-TSSP(β = 1/2) 14 e1 7.468E-5 4.678E-6 2.924E-7 1.868E-8

e2 2.232E-4 1.396E-5 8.732E-7 6.360E-8

18 e1 7.468E-5 4.678E-6 2.924E-7 1.841E-8

e2 2.232E-4 1.396E-5 8.732E-7 5.942E-8

CNFD 14 e1 0.802 3.480E-2 2.855E-2 2.820E-2

e2 0.674 9.012-2 5.005E-2 4.743E-2

18 e1 0.809 1.753E-2 7.363E-3 6.961E-3

e2 0.656 5.491E-2 1.427E-2 1.167E-2

Table 2.2: Time discretization error analysis: e1, e2 at time t=2.

Thirdly, we test the conservation of conserved quantities. Table 2.3 presents the quan-tities and numerical errors at different times with mesh size h = 1

8 and time step k = 0.0001for different numerical methods.

Case II: ‘Subsonic limit’ regime, i.e. 0 < ε 1, we choose B = 1 and C = 1/2εin (8.1), (8.2). Here we test the ε-resolution of different numerical methods. We solvethe problem on the interval [-8, 120], i.e., a = −8 and b = 120 with periodic boundaryconditions. Figure 2.1 shows the numerical results of PSAS-TSSP at t = 1 when we choosethe meshing strategy h = O(ε) and k = O(ε): T0 = (ε0, h0, k0) = (0.125, 0.5, 0.04), T0/4,T0/16; and h = O(ε) and k = 0.04-independent of ε: T0 = (ε0, h0) = (0.125, 0.5), T0/4,T0/16. CN-LF-TSSP with β = 1/4 or β = 1/2 gives similar numerical results at the samemeshing strategies, where CN-LF-TSSP with β = 0 gives correct numerical results atmeshing strategy h = O(ε) and k = O(ε2) and incorrect results at h = O(ε) and k = O(ε)


Time e DGZS PGZS HGZS

PSAS-TSSP 1.0 8.943E-9 3.0000000000 3.41181556 0.5102027362.0 2.360E-8 3.0000000000 3.41181562 0.510202765

β = 0 1.0 7.281E-9 3.0000000000 3.41181557 0.5102027362.0 1.684E-8 3.0000000000 3.41181562 0.510202766

β = 1/4 1.0 1.053E-8 3.0000000000 3.41181556 0.5102027402.0 3.028E-8 3.0000000000 3.41181564 0.510202779

β = 1/2 1.0 1.206E-8 3.0000000000 3.41181556 0.5102027372.0 3.131E-8 3.0000000000 3.41181562 0.510202768

CNFD 1.0 4.745E-3 3.0000000000 3.394829741 0.5101155892.0 8.983E-3 3.0000000000 3.394791238 0.510076710

Table 2.3: Conserved quantities analysis: k = 0.0001 and h = 18 .

[15]. Furthermore, our additional numerical experiments confirm that PSAS-TSSP andCN-LF-TSSP with 1/4 ≤ β ≤ 1/2 are unconditionally stable and CN-LF-TSSP with β = 0is stable under the stability constraint (6.53).

From Tables 2.1-3 and Fig. 2.1, we can draw the following observations:In O(1)-acoustic speed regime, our new methods PSAS-TSSP and CN-LF-TSSP with

β = 1/2 or 1/4 give similar results as the old method, i.e. CN-LF-TSSP with β = 0,proposed in [15]: they are of spectral order accuracy in space discretization and second-order accuracy in time, conserve DGZS exactly and PGZS, HGZS very well (up to 8digits). However, they are improved in two aspects: (i) They are unconditionally stablewhere the old method is conditionally stable under the stability condition k ≤ 2hε

π√d(1−4β)

in d-dimensions (d = 1, 2 or 3); (ii) In the ‘subsonic limit’ regime for initial data with O(ε)-wavelength, i.e. 0 < ε 1, the ε-resolution of our new methods is improved to h = O(ε)and k = O(ε), where the old method required h = O(ε) and k = O(εh) = O(ε2). Thusin the following, we only present numerical results by PSAS-TSSP. In fact, CN-LF-TSSPwith 1/4 ≤ β ≤ 1 gives similar numerical results at the same mesh size and time step forall the following numerical examples.

Example 2.2 Soliton-soliton collisions in one dimension GZS, i.e., we choose d = 1,ε = 1, α = −2 and γ = 0 in (3.40)-(3.41). We use the family of one-soliton solutions(5.5)-(5.7) in [72] to test our new numerical method PSAS-TSSP.

The initial data is chosen as

E(x, 0) = Es(x+ p, 0, η1, V1, ε, ν) + Es(x− p, 0, η2, V2, ε, ν),

N(x, 0) = Ns(x+ p, 0, η1, V1, ε, ν) +Ns(x− p, 0, η2, V2, ε, ν),

∂tN(x, 0) = ∂tNs(x+ p, 0, η1, V1, ε, ν) + ∂tNs(x− p, 0, η2, V2, ε, ν),

where x = ∓p are initial locations of the two solitons.In all the numerical simulations reported in this example, we set λ = 2, and Φ0 = 0. We

only simulated the symmetric collisions, i.e., the collisions of solitons with equal amplitudesη1 = η2 = η and opposite velocities V1 = −V2 ≡ V . Here, we present computations fortwo cases:


a). 0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x b). 0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

c). 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x d).8 10 12 14 16 18 20 22 24

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

e). 58 60 62 64 66 68 700

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x f).58 60 62 64 66 68 70

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

Figure 2.1: Numerical solutions of the electric field |E(x, t)|2 at t = 1 for Example 2.1in the ‘subsonic limit’ regime by PSAS-TSSP. ‘—’: exact solution, ‘+ + +’: numericalsolution. Left column corresponds to h = O(ε) and k = O(ε): a). T0 = (ε0, h0, k0) =(0.125, 0.5, 0.04); c). T0/4; e). T0/16. Right column corresponds to h = O(ε) and k = 0.04-independent ε: b). T0 = (ε0, h0) = (0.125, 0.5); d). T0/4; f). T0/16.

I. Collision between solitons moving with the subsonic velocities, V < 1/ε = 1, i.e. wetake ν = 0.2, η = 0.3 and V = 0.5;

II. Collision between solitons in the transonic regime, V > 1/ε = 1, i.e. we takeν = 2.0, η = 0.3 and V = 3.0.

We solve the problem on the interval [-128,128], i.e., a = -128 and b = 128 with meshsize h = 1

4 and time step k = 0.005. We take p = 10. Figure 2.2 shows the evolution ofthe dispersive wave field |E|2 and the acoustic (nondispersive) field N .


a).

−10

0

10 10

15

20

25

30

0

0.5

1

1.5

t

|E|2

x

−10

0

10 1015

2025

300

0.2

0.4

tx

N

b).−20 −10 0 10 20

0

5

10

15

0

0.5

1

1.5

2

t

x

|E|2

−200

2040

5

10

15

20

−5

0

5

x

t

N

Figure 2.2: Evolution of the wave field |E|2 (left column) and acoustic field N (rightcolumn) in Example 2.2. a). For case I; b). For case II.

Case I corresponds to a soliton-soliton collision when the ratio ν/λ is small, i.e., theGZS (6.1), (6.2) is close to the NLS equation. As is seen, the collision seems quite elastic(cf. Fig. 2.2a). This also validates the formal reduction from GZS to NLS in section 2.5.Case II corresponds to the collision of two transonic solitons. Note that the emission ofthe sound waves is inconspicuous at this value of V (cf. Fig. 2.2b).

From Fig. 2.2, we can see that the unconditionally stable numerical method PSAS-TSSP can really be applied to solve solitary-wave collisions of GZS.

Chapter 3

The Maxwell-Dirac system

3.1 Introduction

One of the fundamental quantum-relativistic equations is given by the Maxwell-Diracsystem (MD), i.e. the Dirac equation [42, 116] for the electron as a spinor coupled tothe Maxwell equations for the electromagnetic field. It represents the time-evolution offast (relativistic) electrons and positrons within self-consistent generated electromagneticfields. In its most compact form, the Dirac equation reads [20, 90]

(ihγη∂η −m0c+ eγηAη)Ψ = 0. (1.1)

Here the unknown Ψ is the 4-vector complex wave function of the “spinorfield”: Ψ(t,x) =(Ψ1,Ψ2,Ψ3,Ψ4)

T ∈ C4, x0 = ct, x = (x1, x2, x3)T ∈ R3 with x0,x denoting the time - resp.

spatial coordinates in Minkowski space. ∂η, stands for ∂∂xη

, i.e. ∂0 = ∂∂x0

= 1c∂∂t , ∂k = ∂

∂xk

(k = 1, 2, 3), where we consequently adopt notation that Greek letter η denotes 0, 1, 2, 3and k denotes the 3 spatial dimension indices 1, 2, 3. γηAη stands for the summation∑3η=0 γ

ηAη. The physical constants are: h for the Planck constant, c for the speed of light,m0 for the electron’s rest mass, and e for the unit charge. By γη ∈ C4×4, η = 0, . . . , 3, wedenote the 4 × 4 Dirac matrices given by

γ0 =

(I2 00 −I2

), γk =

(0 σk

−σk 0

), k = 1, 2, 3,

where Im (m a positive integer) is the m×m identity matrix and σk (k = 1, 2, 3) the 2×2Pauli matrices, i.e.

σ1 :=

(0 11 0

), σ2 :=

(0 −ii 0

), σ3 :=

(1 00 −1

).

Aη(t,x) ∈ R, η = 0, . . . , 3, are the components of the time-dependent electromagnetic po-tential, in particular V (t,x) = −A0(t,x) is the electric potential and A(t,x) = (A1, A2, A3)

T

is the magnetic potential vector. Hence the electric and magnetic fields are given by

E(t,x) = ∇A0 − ∂tA = −∇V − ∂tA, B(t,x) = curl A = ∇× A. (1.2)

79

80 CHAPTER 3. THE MAXWELL-DIRAC SYSTEM

In order to determine the electric and magnetic potentials from fields uniquely, we haveto choose a gauge. In practice, the Lorentz gauge condition is often introduced

L(t,x) :=1

c∂tV + ∇ ·A = −1

c∂tA0 + ∇ ·A = 0. (1.3)

Thus the electric and magnetic fields are governed by the Maxwell equation:

−1

c∂tE + ∇× B =

1

cε0J, ∇ · B = 0, (1.4)

1

c∂tB + ∇× E = 0, ∇ · E =

1

ε0ρ, (1.5)

where ε0 is the permittivity of the free space. The particle density ρ and current densityJ = (j1, j2, j3)

T are defined as follows:

ρ = e|Ψ|2 := e4∑

j=1

|Ψj|2, jk = ec < Ψ, αkΨ >:= ecΨTαkΨ, k = 1, 2, 3, (1.6)

where f denotes the conjugate of f and

αk = γ0γk =

(0 σk

σk 0

), k = 1, 2, 3. (1.7)

From now on, we adopt the standard notations | · |, < ·, · > and ‖ · ‖ for l2-norm of avector, inner product and L2-norm of a function.

Separating the time derivative associated to the “relativistic time variable” x0 = ctand applying γ0 from left of (1.1), plugging (1.2) into (1.4) and (1.5), noticing (1.3), wehave the following Maxwell-Dirac system [108]

ih∂tΨ =3∑

k=1

αk (−ihc∂k − eAk) Ψ + eVΨ +m0c2βΨ, (1.8)

(1

c2∂2t −4

)V =

1

ε0ρ,

(1

c2∂2t −4

)A =

1

cε0J. (1.9)

The vector wave function Ψ is normalized as

‖Ψ(t, ·)‖2 :=

∫

R3|Ψ(t,x)|2 dx = 1. (1.10)

The MD system (1.8), (1.9) represents the time-evolution of fast (relativistic) electronsand positrons within self-consistent generated electromagnetic fields. From the mathemat-ical point of view, the strongly nonlinear MD system poses a hard problem in the study ofPDE’s arising from quantum physics. Well posedness and existence of solutions on all ofR3 but only locally in time, has been proved almost forty years ago [29, 69]. In particularthere are no global existence results without smallness assumptions on the initial data[51, 57]. Thus the MD system is quite involved from the numerical point of view as itposes major open problems from analytical point of view. For solitary solution of MD, werefer [1, 22, 35, 36, 90]

3.2. THE MAXWELL-DIRAC SYSTEM 81

3.2 The Maxwell-Dirac system

Consider the Maxwell-Dirac system represents the time-evolution of fast (relativistic) elec-trons and positrons within external and self-consistent generated electromagnetic fields[108]

ih∂tΨ =3∑

k=1

αk[−ihc∂k − e

(Ak +Aext

k

)]Ψ + e

(V + V ext

)Ψ +m0c

2βΨ, (2.1)

(1

c2∂2t −4

)V =

1

ε0ρ,

(1

c2∂2t −4

)A =

1

cε0J, (2.2)

where V ext = V ext(t,x) ∈ R and Aext(t,x) = (Aext1 , Aext

2 , Aext3 )T ∈ R3 are the external

electric and magnetic potentials respectively.

3.2.1 Dimensionless Maxwell-Dirac system

We rescale the MD (2.1)-(2.2) under the normalization (6.43) by introducing a referencevelocity v, length L = e2/m0v

2ε0, time T = v/L, and strength of the electromagneticpotential λ = e/Lε0, as

x =x

L, t =

t

T, Ψ(t, x) = L3/2Ψ(t,x), V (t, x) = λV (t,x), (2.3)

A(t, x) = λA(t,x), Aext(t, x) = λAext(t,x), V ext(t, x) = λV ext(t,x). (2.4)

Plugging (2.3), (2.4) into (2.1), (2.2), then removing all , we get the following dimensionlessMD:

iδ∂tΨ = −iδε

3∑

k=1

αk∂kΨ −3∑

k=1

αk(Ak +Aextk )Ψ + (V + V ext)Ψ +

1

ε2βΨ, (2.5)

(ε2∂2

t −4)V = ρ,

(ε2∂2

t −4)A = εJ. (2.6)

Two important dimensionless parameters in the MD (2.5), (2.6) are given by the ratio ofthe reference velocity to the speed of light, i.e. ε, and the scaled Planck constant, i.e. δ,as

ε :=v

c, δ :=

hε0v

e2. (2.7)

The position and current densities, Lorentz gauge, as well as electric and magneticfields in dimensionless variables are

ρ(t,x) = |Ψ(t,x)|2, jk(t,x) =1

ε< Ψ(t,x), αkΨ(t,x) >, k = 1, 2, 3, (2.8)

L(t,x) = ε∂tV (t,x) + ∇ ·A(t,x), t ≥ 0, x ∈ R3, (2.9)

E(t,x) = −ε∂tA(t,x) −∇V (t,x), B(t,x) = ∇× A(t,x). (2.10)

When v ≈ c and choosing v = c, then ε = 1 in (2.7) and the MD (2.5), (2.6) collapseto a one-parameter model which is used in [108] to study classical limit and semiclassicalasymptotics of MD. In this case, the parameter δ is the same as the canonical parameterα used in physical literatures [42, 116]. When v c and choosing v = e2/hε0, then δ = 1


and 0 < ε 1 in (2.7), again the MD (2.5), (2.6) collapse to a one-parameter modelwhich is called as ‘nonrelativistic’ limit regime and used in [20, 21, 46, 76, 93, 96] to studysemi-nonrelativistic limits of MD, i.e. letting ε → 0 in (2.5), (2.6). For electrons, ε = 1and δ ≈ 10.9149 [108].

The MD system (2.5)-(2.6) together with initial data

Ψ(0,x) = Ψ(0)(x) with ‖Ψ(0)‖ =

∫

R3|Ψ(0)(x)|2 dx = 1, (2.11)

V (0,x) = V (0)(x), ∂tV (0,x) = V (1)(x), x ∈ R3, (2.12)

A(0,x) = A(0)(x), ∂tA(0,x) = A(1)(x), (2.13)

is time-reversible and time-transverse invariant, i.e., if constants α0 and α1 are added toV (0) and V (1) respectively in (2.12), then the solution V get added by α0 + α1 t and Ψget multiplied by e−it(α0+α1 t/2)/δ , which leaves density of each particle |ψj | (j = 1, 2, 3, 4)unchanged. Moreover, multiplying (2.5) by Ψ and taking imaginary parts we obtain theconservation law:

∂tρ(t,x) + ∇ · J(t,x) = 0, t ≥ 0, x ∈ R3. (2.14)

From (2.14) and (2.6), we get the Lorentz gauge of the MD system (2.5)-(2.6) satisfying

(ε2∂2

t −4)L(t,x) = ε (∂tρ+ ∇ · J) = 0, t ≥ 0, x ∈ R3, (2.15)

L(0,x) = ε∂tV (0,x) + ∇ ·A(0,x) = εV (1)(x) + ∇ · A(0)(x), (2.16)

∂tL(0,x) = ε∂ttV (0,x) + ∇ · ∂tA(0,x) =1

ε[ρ(0,x) + 4V (0,x)] + ∇ · A(1)(x)

=1

ε

[4V (0)(x) + |Ψ(0)(x)|2 + ε∇ ·A(1)(x)

], x ∈ R3. (2.17)

Thus if the initial data in (2.11)-(2.13) satisfies

εV (1)(x) +∇ ·A(0)(x) ≡ 0, 4V (0)(x) + |Ψ(0)(x)|2 + ε∇ ·A(1)(x) ≡ 0, x ∈ R3, (2.18)

which implies

L(0,x) = 0, ∂tL(0,x) = 0, x ∈ R3, (2.19)

the gauge is henceforth conserved during the time-evolution of the MD (2.5)-(2.6).

3.2.2 Plane wave solution

If the initial data in (2.11)-(2.13) for the MD (2.5), (2.6) are chosen as

Ψ(0)(x) = Ψ(0) eiω·x = Ψ(0) ei(ω1x1+ω2x2+ω3x3), (2.20)

V (0)(x) ≡ V (0), V (1)(x) ≡ V (1), x ∈ R3, (2.21)

A(0)(x) ≡ A(0) =

A

(0)1

A(0)2

A(0)3

, A(1)(x) ≡ A(1) =

A

(1)1

A(1)2

A(1)3

, (2.22)

3.3. NUMERICAL METHOD 83

where ω = (ω1, ω2, ω3)T with ωj (j = 1, 2, 3) integers, V (0), V (1) constants, Ψ(0), A(0),

A(1) constant vectors, and

Ψ(0) =1

4π√π√δ−2 + |ω|2 − δ−1

√δ−2 + |ω|2

ω3

ω1 + iω2√δ−2 + |ω|2 − δ−1

0

,

then the MD (2.5)-(2.6) with ε = 1, Aext = −A and V ext = −V , i.e. free MD [37], admitsthe following plane wave solution

Ψ(t,x) = Ψ(0) exp

(iω · x− it

√δ−2 + |ω|2

), (2.23)

V (t,x) = −V ext = V (0) + V (1)t+1

16π3t2, x ∈ R3, t ≥ 0, (2.24)

A(t,x) = −Aext = A(0) + A(1)t+1

2J(0) t2, (2.25)

where J(0) = (j(0)1 , j

(0)2 , j

(0)3 )T and

j(0)k =

ωk

8π3√δ−2 + |ω|2 , k = 1, 2, 3.

Here the normalization condition for the wave function is set as∫ π

−π

∫ π

−π

∫ π

−π|Ψ(t,x)|2 dx = 1.

3.3 Numerical method

In this section we present an explicit, unconditionally stable and accurate numericalmethod for the MD (2.5), (2.6). We shall introduce the method in 3D on a box withperiodic boundary conditions. For 3D in a box Ω = [a1, b1]× [a2, b2]× [a3, b3], the problemwith initial and boundary conditions become

iδ∂tΨ = −iδε

3∑

k=1

αk∂kΨ −3∑

k=1

αk(Ak +Aextk )Ψ + (V + V ext)Ψ +

1

ε2βΨ, (3.1)

(ε2∂2

t −4)V (t,x) = ρ,

(ε2∂2

t −4)A(t,x) = εJ, x ∈ Ω, t > 0, (3.2)

Ψ(0,x) = Ψ(0)(x), V (0,x) = V (0)(x), ∂tV (0,x) = V (1)(x), (3.3)

A(0,x) = A(0)(x), ∂tA(0,x) = A(1)(x), x ∈ Ω, (3.4)

with periodic boundary conditions for Ψ, V,A on ∂Ω, (3.5)

where V (1) and A(0) satisfy

εV (1)(x) + ∇ · A(0)(x) = 0, x ∈ Ω (3.6)

and the normalization condition for the wave function is set as

‖Ψ(t, ·)‖2 :=

∫

Ω|Ψ(t,x)|2 dx =

∫

Ω|Ψ(0)(x)|2 dx = 1. (3.7)


Moreover, integrating the first equation in (3.2) we obtain

ε2d2

dt2

∫

ΩV (t,x) dx =

∫

Ωρ(t,x) dx =

∫

Ω|Ψ(t,x)|2 dx = 1, t ≥ 0. (3.8)

This implies that

Mean(V (t, ·)) = Mean(V (0)) + t Mean(V (1)) +t2

2ε2, t ≥ 0, (3.9)

where

Mean(f) :=

∫

Ωf(x) dx.

3.3.1 Time-splitting spectral discretization

We choose the spatial mesh size hj =bj−aj

Mj(j = 1, 2, 3) in xj-direction with Mj given

integer and time step ∆t. Denote the grid points as

xp,q,r = (x1,p, x2,q, x3,r)T , (p, q, r) ∈ N ,

where

N = (p, q, r) | 0 ≤ p ≤M1, 0 ≤ q ≤M2, 0 ≤ r ≤M3 ,x1,p = a1 + ph1, x2,q = a2 + qh2, x3,r = a3 + rh3, (p, q, r) ∈ N

and time step as

tn = n ∆t, tn+1/2 = (n+ 1/2)∆t, n = 0, 1, · · ·

Let Ψnp,q,r, V

np,q,r and An

p,q,r be the numerical approximation of Ψ(tn,xp,q,r), V (tn,xp,q,r)and A(tn,xp,q,r) respectively. Furthermore let Ψn, V n and An be the solution vector attime t = tn with components Ψn

p,q,r, Vnp,q,r and An

p,q,r respectively.From time t = tn to t = tn+1, we discretize the MD (3.1)-(3.2) as follows: The

nonlinear wave-type equations (3.2) are discretized by pseudospectral method for spatialderivatives and then solving the ODEs in phase space analytically under appropriatechosen transmission conditions between different time intervals, and the Dirac equation(3.1) is solved in two splitting steps. For the nonlinear wave-type equations (3.2), weassume

V (t,x) =∑

(j,k,l)∈M

V nj,k,l(t) e

iµj,k,l·(x−a), x ∈ Ω, tn ≤ t ≤ tn+1, (3.10)

where f denotes the Fourier coefficients of f and

M =

(j, k, l) | − M1

2≤ j <

M1

2, −M2

2≤ k <

M2

2, −M3

2≤ l <

M3

2

,

µj,k,l =

µ

(1)j

µ(2)k

µ(3)l

, a =

a1

a2

a3

,


with

µ(1)j =

2πj

b1 − a1, µ

(2)k =

2πk

b2 − a2, µ

(3)l =

2πl

b3 − a3, (j, k, l) ∈ M.

Plugging (6.28) and (2.8) into (3.2), noticing the orthogonality of the Fourier series, weget the following ODEs for n ≥ 0:

ε2d2V n

j,k,l(t)

d t2+ |µj,k,l|2 V n

j,k,l(t) = ρj,k,l(tn) := (|Ψn|2)j,k,l, tn ≤ t ≤ tn+1, (3.11)

V nj,k,l(tn) =

(V (0))j,k,l, n = 0,

V n−1j,k,l (tn), n > 0,

(j, k, l) ∈ M. (3.12)

As noticed in [14], for each fixed (j, k, l) ∈ M, Eq. (6.29) is a second-order ODE. Itneeds two initial conditions such that the solution is unique. When n = 0 in (6.29), (6.30),we have the initial condition (6.30) and we can pose the other initial condition for (6.29)due to the initial condition (3.3) for the MD (3.1)-(3.2):

d

dtV 0j,k,l(t0) =

d

dtV 0j,k,l(0) = (V (1))j,k,l. (3.13)

Then the solution of (6.29), (6.30) and (6.31) for t ∈ [0, t1] is:

V 0j,k,l(t) =

(V (0))j,k,l + t (V (1))j,k,l +˜(|Ψ(0)|2)j,k,l t2/2ε2 j = k = l = 0,

[(V (0))j,k,l − ˜(|Ψ(0)|2)j,k,l/|µj,k,l|2

]cos(t|µj,k,l|/ε) otherwise.

+(V (1))j,k,l sin(t|µj,k,l|/ε)ε

|µj,k,l|+ ˜(|Ψ(0)|2)j,k,l/|µj,k,l|2

But when n > 0, we only have one initial condition (6.30). One can’t simply pose thecontinuity between d

dt Vnj,k,l(t) and d

dt Vn−1j,k,l (t) across the time t = tn due to the right hand

side in (6.29) is usually different in two adjacent time intervals [tn−1, tn] and [tn, tn+1], i.e.

ρj,k,l(tn−1) = ˜(|Ψn−1|2)j,k,l 6= (|Ψn|2)j,k,l = ρj,k,l(tn). Since our goal is to develop explicitscheme and we need linearize the nonlinear term in (3.2) in our discretization (6.29), ingeneral,

d

dtV n−1j,k,l (t−n ) = lim

t→t−n

d

dtV n−1j,k,l (t) 6= lim

t→t+n

d

dtV nj,k,l(t) =

d

dtV nj,k,l(t

+m),

n = 1, · · · , (j, k, l) ∈ M. (3.14)

Unfortunately, we don’t know the jump ddt V

nj,k,l(t

+n ) − d

dt Vn−1j,k,l (t

−n ) across the time t = tn.

In order to get a unique solution of (6.29), (6.30) for n > 0, here we pose an additionalcondition:

V nj,k,l(tn−1) = V n−1

j,k,l (tn−1), (j, k, l) ∈ M. (3.15)

The condition (6.34) is equivalent to pose the solution V nj,k,l(t) on the time interval [tn, tn+1]

of (6.29), (6.30) is also continuity at the time t = tn−1. After a simple computation, we


get the solution of (6.29), (6.30) and (6.34) for n > 0:

V nj,k,l(t) =

V nj,k,l(tn) + ρj,k,l(tn)(t− tn)

2/2ε2+ j = k = l = 0,t−tn∆t

[V nj,k,l(tn) − V n−1

j,k,l (tn−1) + ρj,k,l(tn)(∆t)2/2ε2

]

[V nj,k,l − ρj,k,l(tn)/|µj,k,l|2

]cos((t− tn)|µj,k,l|/ε)+ otherwise.[

(1 − cos(|µj,k,l|∆t/ε))ρj,k,l(tn)/|µj,k,l|2 − V n−1j,k,l (tn−1)

+V nj,k,l(tn) cos(|µj,k,l|∆t/ε)

] sin((t− tn)|µj,k,l|/ε)sin(|µj,k,l|∆t/ε)

+ρj,k,l(tn)/|µj,k,l|2

Discretization for the equation of A in (3.2) can be done in a similar way.

For the Dirac equation (3.1), we solve it in two splitting steps. One solves first

iδ∂tΨ(t,x) = −iδε

3∑

k=1

αk∂kΨ +1

ε2βΨ, x ∈ Ω, tn ≤ t ≤ tn+1, (3.16)

for the time step of length ∆t, followed by solving

iδ∂tΨ(t,x) = (V + V ext)Ψ −3∑

k=1

αk(Ak +Aextk )Ψ = G(t,x) Ψ, (3.17)

for the same time step with

G(t,x) =

[(V (t,x) + V ext(t,x)

)I4 −

3∑

k=1

αk(Ak(t,x) +Aext

k (t,x))]. (3.18)

For each fixed x ∈ Ω, integrating (11.15) from tn to tn+1, and then approximating theintegral on [tn, tn+1] via the Simpson rule, one reads

Ψ(tn+1,x) = exp

[−i1δ

∫ tn+1

tnG(t,x) dt

]Ψ(tn,x)

≈ exp

[−i∆t

δ

(G(tn,x) + 4G(tn+1/2,x) +G(tn+1,x)

)/6

]Ψ(tn,x)

= exp

[−i∆t

δGn+1/2(x)

]Ψ(tn,x). (3.19)

Since Gn+1/2(x) is a U-matrix, i.e. (Gn+1/2(x))T = Gn+1/2(x), it is diagonalizable (seedetail in Remark 3.3.2), i.e. there exist a diagonal matrix Dn+1/2(x) and a complexorthogonormal matrix Pn+1/2(x), i.e. (Pn+1/2(x))T = (Pn+1/2(x))−1, such that

Gn+1/2(x) = Pn+1/2(x) Dn+1/2(x) (Pn+1/2(x))T , x ∈ Ω. (3.20)

Plugging (3.20) into (3.19), we obtain

Ψ(tn+1,x) = Pn+1/2(x) exp

[−i∆t

δDn+1/2(x)

](Pn+1/2(x))T Ψ(tn,x), x ∈ Ω. (3.21)


For discretizing (11.8), we assume

Ψ(t,x) =∑

(j,k,l)∈M

Ψj,k,l(t) eiµj,k,l·(x−a), x ∈ Ω, tn ≤ t ≤ tn+1. (3.22)

Substituting (3.22) into (11.8), we have

dΨj,k,l(t)

dt= − i

εMj,k,l Ψj,k,l(t), tn ≤ t ≤ tn+1, (j, k, l) ∈ M, (3.23)

where the matrix

Mj,k,l = µ(1)j α1 + µ

(2)k α2 + µ

(3)l α3 + ε−1δ−1. (3.24)

Since Mj,k,l is a U-matrix, again it is diagonalizable (see detail in Remark 3.3.3), i.e. thereexist a diagonal matrix Dj,k,l and a complex orthogonormal matrix Pj,k,l such that

Mj,k,l = Pj,k,l Dj,k,l (Pj,k,l)T , (j, k, l) ∈ M. (3.25)

Thus the solution of (3.23) is

Ψj,k,l(t) = exp

[− i

ε(t− tn)Mj,k,l

]Ψj,k,l(tn)

= Pj,k,l exp

[− i

ε(t− tn)Dj,k,l

](Pj,k,l)

T Ψj,k,l(tn), tn ≤ t ≤ tn+1. (3.26)

From time t = tn to t = tn+1, we combine the splitting steps via the standard Strangsplitting:

V n+1p,q,r =

∑

(j,k,l)∈M

V nj,k,l(tn+1) e

iµj,k,l·(xp,q,r−a), (3.27)

An+1p,q,r =

∑

(j,k,l)∈M

Anj,k,l(tn+1) e

iµj,k,l·(xp,q,r−a), (p, q, r) ∈ N , (3.28)

Ψ∗p,q,r =

∑

(j,k,l)∈M

Pj,k,l exp

[− i∆t

2εDj,k,l

](Pj,k,l)

T (Ψn)j,k,l eiµj,k,l·(xp,q,r−a),

Ψ∗∗p,q,r = Pn+1/2(xp,q,r) exp

[−i∆t

δDn+1/2(xp,q,r)

](Pn+1/2(xp,q,r))

T Ψ∗p,q,r,

Ψn+1p,q,r =

∑

(j,k,l)∈M

Pj,k,l exp

[− i∆t

2εDj,k,l

](Pj,k,l)

T (Ψ∗∗)j,k,l eiµj,k,l·(xp,q,r−a), (3.29)

where the formula for V nj,k,l(tn+1) and An

j,k,l(tn+1) are given in Remark 3.3.4, and Uj,k,lthe discrete Fourier transform coefficients of the vector Up,q,r, (p, q, r) ∈ N, are definedas

Uj,k,l =1

M1M2M3

∑

(p,q,r)∈Q

Up,q,r eiµj,k,l·(xp,q,r−a), (j, k, l) ∈ M, (3.30)

where

Q = (p, q, r) | 0 ≤ p ≤M1 − 1, 0 ≤ q ≤M2 − 1, 0 ≤ r ≤M3 − 1 .


The initial conditions (3.3), (3.4) are discretized as

Ψ0p,q,r = Ψ(0)(xp,q,r), V 0

p,q,r = V (0)(xp,q,r),dV 0

p,q,r(0)

dt= V (1)(xp,q,r),

A0p,q,r = A(0)(xp,q,r),

dA0p,q,r(0)

dt= A(1)(xp,q,r), (p, q, r) ∈ N .

Remark 3.3.1 We use the Simpson quadrature rule to approximate the integration in(3.19) instead of the trapezodial rule which was used in [14, 15] for a similar integration.The reason is that we want the quadrature is exact when the MD system (2.5)-(2.6) admitsthe plane wave solution (2.23)-(2.25). In this case, the integrand G(t,x) is quadratic int. Thus the algorithm (3.27)-(3.29) gives exact results when the MD system admits planewave solution.

Remark 3.3.2 Diagnolize the matrix Gn+1/2(x) in (3.19) and computation:From (3.19), notice (3.18), we have

Gn+1/2(x) =1

6

[G(tn,x) + 4G(tn+1/2,x) +G(tn+1,x)

]

=

V n+1/2(x) 0 −An+1/23 (x) −An+1/2

− (x)

0 V n+1/2(x) −An+1/2+ (x) A

n+1/23 (x)

−An+1/23 (x) −An+1/2

− (x) V n+1/2(x) 0

−An+1/2+ (x) A

n+1/23 (x) 0 V n+1/2(x)

, (3.31)

with

An+1/2± (x) = A

n+1/21 (x) ± iA

n+1/22 (x),

V n+1/2(x) =1

6

[V (tn,x) + V ext(tn,x) + 4(V (tn+1/2,x) + V ext(tn+1/2,x))

+V (tn+1,x) + V ext(tn+1,x)],

An+1/2(x) =(An+1/21 (x), A

n+1/22 (x), A

n+1/23 (x)

)T, x ∈ Ω,

An+1/2k (x) =

1

6

[Ak(tn,x) +Aext

k (tn,x) + 4(Ak(tn+1/2,x) +Aextk (tn+1/2,x))

+Ak(tn+1,x) +Aextk (tn+1,x)

], k = 1, 2, 3.

Since Gn+1/2(x) is a U -matrix, it is diagonalizable. The characteristic polynomial ofGn+1/2(x) is

det(λI4 −Gn+1/2(x)

)

=

∣∣∣∣∣∣∣∣∣∣

λ− V n+1/2(x) 0 An+1/23 (x) A

n+1/2− (x)

0 λ− V n+1/2(x) An+1/2+ (x) −An+1/2

3 (x)

An+1/23 (x) A

n+1/2− (x) λ− V n+1/2(x) 0

An+1/2+ (x) −An+1/2

3 (x) 0 λ− V n+1/2(x)

∣∣∣∣∣∣∣∣∣∣

=

[(λ− V n+1/2(x)

)2− |An+1/2(x)|2

]2= 0. (3.32)


Thus the eigenvalues of Gn+1/2(x) are:

λn+1/2+ (x), λ

n+1/2+ (x), λ

n+1/2− (x), λ

n+1/2− (x),

with

λn+1/2± (x) = V n+1/2(x) ± |An+1/2(x)| = V n+1/2(x) ±

√√√√3∑

j=1

|An+1/2j (x)|2

and the corresponding eigenvectors are:

vn+1/21 (x) =

An+1/2− (x)

−An+1/23 (x)

0

|An+1/2(x)|

, v

n+1/22 (x) =

An+1/23 (x)

An+1/2+ (x)

|An+1/2(x)|0

,

vn+1/23 (x) =

0

−|An+1/2(x)|An+1/2− (x)

−An+1/23 (x)

, v

n+1/24 (x) =

−|An+1/2(x)|0

An+1/23 (x)

An+1/2+ (x)

,

Let

Dn+1/2(x) = diag(λn+1/2+ (x), λ

n+1/2+ (x), λ

n+1/2− (x), λ

n+1/2− (x)

),

Pn+1/2(x) =1√

2|An+1/2(x)|(vn+1/21 (x) v

n+1/22 (x) v

n+1/23 (x) v

n+1/24 (x)

).

Thus Dn+1/2(x) is a diagonal matrix, Pn+1/2(x) is a complex orthogonormal matrix, andthey diagonalize the matrix Gn+1/2(x), i.e.

Gn+1/2(x) = Pn+1/2(x) Dn+1/2(x) (Pn+1/2(x))T , x ∈ Ω. (3.33)

In order to compute Gn+1/2(xp,q,r) ((p, q, r) ∈ M) used in (3.31), we need V (tn,xp,q,r) =V np,q,r, V (tn+1,xp,q,r) = V n+1

p,q,r , A(tn,xp,q,r) = Anp,q,r, A(tn+1,xp,q,r) = An+1

p,q,r, V (tn+1/2,xp,q,r)and A(tn+1/2,xp,q,r). The first four terms are given in (3.27) and (3.28). The last twoterms can be computed as following:

V (tn+1/2,xp,q,r) =∑

(j,k,l)∈M

V nj,k,l(tn+1/2) e

iµj,k,l·(xp,q,r−a),

A(tn+1/2,xp,q,r) =∑

(j,k,l)∈M

Anj,k,l(tn+1/2) e

iµj,k,l·(xp,q,r−a),

where for n = 0:

V 0j,k,l(t1/2) =

(V (0))j,k,l +∆t2 (V (1))j,k,l +

˜(|Ψ(0)|2)j,k,l (∆t)2/8ε2, j = k = l = 0,

[(V (0))j,k,l − ˜(|Ψ(0)|2)j,k,l/|µj,k,l|2

]cos(∆t|µj,k,l|/2ε) otherwise.

+(V (1))j,k,l sin(∆t |µj,k,l|/2ε)ε

|µj,k,l|+ ˜(|Ψ(0)|2)j,k,l/|µj,k,l|2


=

∣∣∣∣∣∣∣∣∣∣∣

λ− ε−1δ−1 0 −µ(3)l −µ(1)

j + iµ(2)k

0 λ− ε−1δ−1 −µ(1)j − iµ

(2)k µ

(3)l

−µ(3)l −µ(1)

j + iµ(2)k λ+ ε−1δ−1 0

−µ(1)j − iµ

(2)k µ

(3)l 0 λ+ ε−1δ−1

∣∣∣∣∣∣∣∣∣∣∣

=(λ2 − ε−2δ−2 − |µj,k,l|2

)2= 0 (3.34)

Thus the eigenvalues of Mj,k,l are:

λj,k,l, λj,k,l, −λj,k,l, −λj,k,l with λj,k,l =√ε−2δ−2 + |µj,k,l|2

and the corresponding eigenvectors are:

v1j,k,l =

λj,k,l + ε−1δ−1

0

µ(3)l

µ(1)j + iµ

(2)k

, v2

j,k,l =

0λj,k,l + ε−1δ−1

µ(1)j − iµ

(2)k

−µ(3)l

,

v3j,k,l =

−µ(3)l

−µ(1)j − iµ

(2)k

λj,k,l + ε−1δ−1

0

, v4

j,k,l =

−µ(1)j + iµ

(2)k

µ(3)l

0λj,k,l + ε−1δ−1

.

Let

Dj,k,l = diag (λj,k,l, λj,k,l,−λj,k,l,−λj,k,l) ,

Pj,k,l =1√

2(λ2j,k,l + ε−1δ−1λj,k,l)

(v1j,k,l v2

j,k,l v3j,k,l v4

j,k,l

).

Thus Dj,k,l is a diagonal matrix, Pj,k,l is a complex orthogonormal matrix, and they diag-onalize the matrix Mj,k,l, i.e.

Mj,k,l = Pj,k,l Dj,k,l (Pj,k,l)T , (j, k, l) ∈ M. (3.35)

Remark 3.3.4 Computation of V nj,k,l(tn+1) and V n

j,k,l(tn+1) in (3.27&28):

For n = 0:

V 0j,k,l(t1) =

(V (0))j,k,l + ∆t (V (1))j,k,l +˜(|Ψ(0)|2)j,k,l (∆t)2/2ε2, j = k = l = 0,

[(V (0))j,k,l − ˜(|Ψ(0)|2)j,k,l/|µj,k,l|2

]cos(∆t|µj,k,l|/ε) otherwise.

+(V (1))j,k,l sin(∆t |µj,k,l|/ε)ε

|µj,k,l|+ ˜(|Ψ(0)|2)j,k,l/|µj,k,l|2

(3.36)


A0j,k,l(t1) =

(A(0))j,k,l + ∆t (A(1))j,k,l + (J(0))j,k,l (∆t)2/2ε2, j = k = l = 0,

[(A(0))j,k,l − (J(0))j,k,l/|µj,k,l|2


+(A(1))j,k,l sin(∆t |µj,k,l|/ε)ε

|µj,k,l|+(J(0))j,k,l/|µj,k,l|2

(3.37)

and for n > 0:

V nj,k,l(tn+1) =

2V nj,k,l(tn) − V n−1

j,k,l (tn−1) + (|Ψn|2)j,k,l(∆t)2/ε2 j = k = l = 0,

2

[V nj,k,l − (|Ψn|2)j,k,l/|µj,k,l|2


−V n−1j,k,l (tn−1) + 2(|Ψn|2)j,k,l/|µj,k,l|2

(3.38)

Anj,k,l(tn+1) =

2Anj,k,l(tn) − An−1

j,k,l(tn−1) + (J(n))j,k,l(∆t)2/ε2 j = k = l = 0,

2

[Anj,k,l − (J(n))j,k,l/|µj,k,l|2


−An−1j,k,l(tn−1) + 2(J(n))j,k,l/|µj,k,l|2

(3.39)

3.3.2 Properties of the numerical method

(1). Plane wave solution: If the initial data in (3.3), (3.4) are chosen as in (2.20)-(2.22)and the external electric and magnetic fields, i.e. V ext and Aext, are chosen as in (2.24)-(2.25), the MD system (3.1)-(3.5) admits the plane wave solution (2.23)-(2.25). It is easyto see that in this case our numerical method (3.27)-(3.29) gives exact results providedthat Mj ≥ 2 (|ωj| + 1) (j = 1, 2, 3).

(2). Time transverse invariant: If constants α0 and α1 are added to V (0) and V (1)

respectively in (3.3), then the solution V n get added by α0 +α1 tn and Ψn get multipliedby e−itn(α0+α1 tn/2)/δ , which leaves density of each particle |ψnj | (j = 1, 2, 3, 4) unchanged.

(3). Conservation: Let U = Up,q,r, (p, q, r) ∈ N and f(x) a periodic function on thebox Ω, and let ‖ · ‖l2 be the usual discrete l2-norm on the box Ω, i.e.

‖U‖2l2 = h1h2h3

∑

(p,q,r)∈Q

|Up,q,r|2, (3.40)

DMean(U) = h1h2h3

∑

(p,q,r)∈Q

Up,q,r, (3.41)

‖f‖2l2 = h1h2h3

∑

(p,q,r)∈Q

|f(xp,q,r)|2. (3.42)

Then we have


Theorem 3.3.1 The time splitting spectral method (3.27)-(3.29) for the MD conservesthe following quantities in the discretized level:

‖Ψn‖l2 = ‖Ψ0‖l2 = ‖Ψ(0)‖l2 , n = 0, 1, 2, · · · , (3.43)

DMean(V n) = DMean(V (0)) + tn DMean(V (1)) +t2n2ε2

DMean(|Ψ(0)|2). (3.44)

Proof: From (3.29), notice (3.30), (3.20) and (3.25), Parseval’s equality, we have

1

h1h2h3‖Ψn+1‖2

l2 =∑

(p,q,r)∈Q

∣∣∣Ψn+1p,q,r

∣∣∣2

=∑

(p,q,r)∈Q

∣∣∣∣∣∣∑

(j,k,l)∈M

Pj,k,l exp

[− i∆t

2εDj,k,l

](Pj,k,l)

T (Ψ∗∗)j,k,leiµj,k,l·(xp,q,r−a)

∣∣∣∣∣∣

2

= M1M2M3

∑

(j,k,l)∈M

∣∣∣∣Pj,k,l exp

[− i∆t

2εDj,k,l

](Pj,k,l)

T (Ψ∗∗)j,k,l

∣∣∣∣2

= M1M2M3

∑

(j,k,l)∈M

∣∣∣(Ψ∗∗)j,k,l

∣∣∣2

=1

M1M2M3

∑

(j,k,l)∈M

∣∣∣∣∣∣∑

(p,q,r)∈Q

Ψ∗∗p,q,r e

iµj,k,l·(xp,q,r−a)

∣∣∣∣∣∣

2

=∑

(p,q,r)∈Q

|Ψ∗∗|2

=∑

(p,q,r)∈Q

∣∣∣∣Pn+1/2(xp,q,r) exp

[−i∆t

δDn+1/2(xp,q,r)

](Pn+1/2(xp,q,r))

T Ψ∗p,q,r

∣∣∣∣2

=∑

(p,q,r)∈Q

∣∣∣Ψ∗p,q,r

∣∣∣2

=∑

(p,q,r)∈Q

∣∣∣Ψnp,q,r

∣∣∣2

=1

h1h2h3‖Ψn‖2

l2 , n ≥ 0. (3.45)

Thus the equality (3.43) is obtained by induction.From (3.27), notice (3.36), (3.38), (3.30), we have

V n0,0,0(tn+1) = 2V n

0,0,0(tn) − V n−10,0,0 (tn−1) + (|Ψn|2)0,0,0(∆t)2/ε2

= 2V n0,0,0(tn) − V n−1

0,0,0 (tn−1) +(∆t)2

M1M2M3ε2

∑

(p,q,r)∈Q

|Ψ(0)p,q,r|2

= 2V n−10,0,0 (tn) − V n−1

0,0,0 (tn−1) +(∆t)2

M1M2M3ε2

∑

(p,q,r)∈Q

|Ψ(0)p,q,r|2

= 3V n−20,0,0 (tn−1) − 2V n−2

0,0,0 (tn−2) +(1 + 2)(∆t)2

M1M2M3ε2

∑

(p,q,r)∈Q

|Ψ(0)p,q,r|2 . (3.46)

By induction, we get

V n0,0,0(tn+1) = (n+ 1)V 0

0,0,0(t1) − nV 00,0,0(t0) +

n(n+ 1)(∆t)2

2M1M2M3ε2

∑

(p,q,r)∈Q

|Ψ(0)p,q,r|2


= V(0)0,0,0(t0) + tn+1V

(1)0,0,0(t0) +

t2n+1

2M1M2M3ε2

∑

(p,q,r)∈Q

|Ψ(0)p,q,r|2, n ≥ 0. (3.47)

From (3.27), notice (3.47), (3.30) and (3.41), we get

1

h1h2h3DMean(V n+1) =

∑

(p,q,r)∈Q

V n+1p,q,r

=∑

(p,q,r)∈Q

∑

(j,k,l)∈M

V nj,k,l(tn+1)e

iµj,k,l·(xp,q,r−a)

=∑

(j,k,l)∈M

V nj,k,l(tn+1)

∑

(p,q,r)∈Q

eiµj,k,l·(xp,q,r−a) = M1M2M3Vn0,0,0(tn+1)

= M1M2M3

[V 0

0,0,0(t0) + tn+1V(1)0,0,0(t0)

]+t2n+1

2ε2

∑

(p,q,r)∈Q

|Ψ(0)p,q,r|2

=∑

(p,q,r)∈Q

V (0)p,q,r + tn+1

∑

(p,q,r)∈Q

V (1)p,q,r +

t2n+1

2ε2

∑

(p,q,r)∈Q

|Ψ(0)p,q,r|2, n ≥ 0. (3.48)

Thus the desired equality (3.44) is a combination of (3.41) and (3.48).

(4). Unconditional stability: By the standard Von Neumann analysis for (3.27)-(3.28),noting (3.43), we get the method (3.27)-(3.29) is unconditionally stable. In fact, settingΨn = 0 and plugging V n

j,k,l(tn+1) = µV nj,k,l(tn) = µ2V n−1

j,k,l (tn−1) into (3.38) with |µ| theamplification factor, we obtain the characteristic equation:

µ2 − 2µ cos(|µj,k,l|∆t/ε) + 1 = 0, (j, k, l) ∈ M. (3.49)

This implies

µ = cos(|µj,k,l|∆t/ε) ± i sin(|µj,k,l|∆t/ε). (3.50)

Thus the amplification factor

Gj,k,l = |µ| =√

cos2(|µj,k,l|∆t/ε) + sin2(|µj,k,l|∆t/ε) = 1, (j, k, l) ∈ M. (3.51)

Similar results can be obtained for (3.28). These together with (3.43) imply the method(3.27)-(3.29) is unconditionally stable. This is confirmed by our numerical results in thenext section.

(5). ε-resolution in the ‘nonrelatistic’ limit regime (0 < ε 1): As our nu-merical results in the next section suggest: The meshing strategy (or ε-resolution) whichguarantees ‘good’ numerical results in the ‘nonrelatistic’ limit regime for initial data withO(ε)-wavelength, i.e. 0 < ε 1, is:

h = maxh1, h2, h3 = O(ε), ∆t = O(ε2). (3.52)


3.3.3 For homogeneous Dirichlet boundary conditions

In some cases, the periodic boundary conditions (3.5) may be replaced by the followinghomogeneous Dirichlet boundary conditions:

Ψ(t,x) = V (t,x) = 0, A(t,x) = 0, x ∈ ∂Ω, t ≥ 0. (3.53)

In this case, the method designed above is still valid provided that we replace the Fourierbasis functions by sine basis functions. Let

M = (j, k, l) | 1 ≤ j ≤M1 − 1, 1 ≤ k ≤M2 − 1, 1 ≤ l ≤M3 − 1 ,

µ(1)j =

πj

b1 − a1, µ

(2)k =

πk

b2 − a2, µ

(3)l =

πl

b3 − a3, (j, k, l) ∈ M. (3.54)

The detailed scheme is:

V n+1p,q,r =

∑

(j,k,l)∈M

V nj,k,l(tn+1) sin

(pjπ

M1

)sin

(qkπ

M2

)sin

(rlπ

M3

),

An+1p,q,r =

∑

(j,k,l)∈M

Anj,k,l(tn+1) sin

(pjπ

M1

)sin

(qkπ

M2

)sin

(rlπ

M3

), (p, q, r) ∈ M,

Ψ∗p,q,r =

∑

(j,k,l)∈M

Pj,k,l exp

[− i∆t

2εDj,k,l

](Pj,k,l)

T (Ψn)j,k,l sin

(pjπ

M1

)sin

(qkπ

M2

)sin

(rlπ

M3

),

Ψ∗∗p,q,r = Pn+1/2(xp,q,r) exp

[−i∆t

δDn+1/2(xp,q,r)

](Pn+1/2(xp,q,r))

T Ψ∗p,q,r,

Ψn+1p,q,r =

∑

(j,k,l)∈M

Pj,k,l exp

[− i∆t

2εDj,k,l

](Pj,k,l)

T (Ψ∗∗)j,k,l sin

(pjπ

M1

)sin

(qkπ

M2

)sin

(rlπ

M3

),

where the formula for V nj,k,l(tn+1) and An

j,k,l(tn+1) are given in Remark 3.3.4 with µj,k,l

is replaced by (3.54), and Uj,k,l the discrete sine transform coefficients of the vectorUp,q,r, (p, q, r) ∈ N, are defined as

Uj,k,l =8

M1M2M3

∑

(p,q,r)∈M

Up,q,r sin

(pjπ

M1

)sin

(qkπ

M2

)sin

(rlπ

M3

), (j, k, l) ∈ M. (3.55)

3.4 Numerical results

In this section, we present numerical results to demonstrate spatial/temporal accuracyand meshing strategy of the numerical method for MD. The initial data in (3.3)-(3.4) arechosen as

ψ(0)j (x) =

(γ1γ2γ3)1/4

2π3/4exp[−(γ1x

21 + γ2x

22 + γ3x

23)/2) exp(icjx1/ε), (4.1)

V (0)(x) = 0, V (1)(x) = 0, A(0)(x) = 0, A(1)(x) = 0, x ∈ R3. (4.2)

They decay to zero sufficient fast as |x| → ∞. This Gaussian-type initial data is oftenused to study wave motion and interaction in physical literatures. We always computeon a box, which is large enough such that the periodic boundary conditions (3.5) do not


introduce a significant aliasing error relative to the problem in the whole space. In ourcomputations, we always choose uniform mesh, i.e. h = h1 = h2 = h3.

Example 3.1 Accuracy test and meshing strategy, i.e. we choose δ = 1, V ext(t,x) ≡ 0,Aext(t,x) ≡ 0 in (3.1), γ1 = γ2 = γ3 = 5 and c1 = c2 = c3 = c4 = 1 in (4.1).

We solve MD (3.1)-(3.5) on a box Ω = [−4, 4]3 by using the numerical method (3.27)-(3.29), and present results for two different regimes of velocity, i.e. 1/ε:

Case I. O(1)-velocity speed, i.e. we choose ε = 1 in (3.1), (3.2) and (4.1). Here we testthe spatial and temporal discretization errors. Let Ψ, V and A be the ‘exact’ solutionswhich are obtained numerically by using our numerical method with a very fine mesh andtime step, e.g. h = 1

8 and ∆t = 0.0001, and Ψh,∆t, V h,∆t and Ah,∆t be the numericalsolution obtained by using our method with mesh size h and time step ∆t. To quantifythe numerical method, we define the error functions as

eΨ(t) = ||Ψ(t, ·) − Ψh,∆t(t, ·)||l2 , eA(t) = ||A(t, ·) − Ah,∆t(t, ·)||l2 ,eV (t) = ||V (t, ·) − V h,∆t(t, ·)||l2 .

First, we test the discretization error in space. In order to do this, we choose a verysmall time step, e.g. ∆t = 0.0001, such that the error from time discretization is negligiblecomparing to the spatial discretization error. Table 3.1 lists the numerical errors of eΨ(t),eV (t) and eA(t) at t = 0.4 with different mesh sizes h.

Table 3.1: Spatial discretization error analysis: At time t = 0.4 under ∆t = 0.0001mesh h = 1 h = 0.5 h = 0.25 h = 0.125

eΨ(t) 0.76250 5.8928E − 2 7.7029E − 6 2.2164E − 11eV (t) 6.4937E − 3 3.9210E − 4 8.8440E − 5 7.6842E − 13eA(t) 7.0499E − 3 4.6114E − 4 1.1609E − 6 7.6508E − 13

Secondly, we test the discretization error in time. Table 3.2 shows the numerical errorsof eΨ(t), eV (t) and eA(t) at t = 0.4 under different time step ∆t and mesh size h = 1/4.

Table 3.2: Temporal discretization error analysis: At time t = 0.4 under h = 1/4time step ∆t = 0.05 ∆t = 0.025 ∆t = 0.0125 ∆t = 0.00625

eΨ(t) 7.7643E − 5 1.9592E − 5 4.9041E − 6 1.2063E − 6eV (t) 2.9906E − 4 7.4114E − 5 1.8405E − 5 4.5103E − 6eA(t) 3.5809E − 4 8.8633E − 5 2.2004E − 5 5.381E − 6

Thirdly, we test the density conservation in (3.43). Table 3.3 shows ‖Ψ‖l2 at differenttimes.

Case II: ‘nonrelativistic’ limit regime, i.e. 0 < ε << 1. Here we test the ε-resolution ofour numerical method. Figure 3.1 shows the numerical results at t = 0.4 when we choose


Table 3.3: Density conservation testtime t = 0 t = 1 t = 2

‖Ψ(t, ·)‖l2 0.9999999999999 0.9999999999998 0.9999999999996

a).−4 −2 0 2 4

0

0.05

0.1

0.15

0.2

0.25

x1

|ψ1(x

1,0,0

)|2

d).−4 −2 0 2 4

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x1

A 1(x1,0

,0)

b).−4 −2 0 2 4

0

0.05

0.1

0.15

0.2

0.25

x1

|ψ1(x

1,0,0

)|2

e).−4 −2 0 2 4

0

0.005

0.01

0.015

0.02

0.025

0.03

x1

A 1(x1,0

,0)

c).−4 −2 0 2 4

0

0.05

0.1

0.15

0.2

0.25

x1

|ψ1(x

1,0,0

)|2

f).−4 −2 0 2 4

0

0.005

0.01

0.015

0.02

0.025

x1

A 1(x1,0

,0)

Figure 3.1: Meshing strategy test in Example 3.1 for wave fuction |Ψ1(t, x1, 0, 0)|2 (Leftcolumn) and magnetic popential A1(t, x1, 0, 0) (Right column) at time t = 0.4. ‘—’: ‘exact’solutions, ‘+ + +’: numerical solutions. a). ε = 1, h = 1 and ∆t = 0.2; b). ε = 1/2,h = 1/2 and ∆t = 0.05; c). ε = 1/4, h = 1/4 and ∆t = 0.0125; which corresponds tomeshing strategy: h = O(ε) and ∆t = O(ε2).


the meshing strategy: ε = 1, h = 1/2, ∆t = 0.2; ε = 1/2, h = 1/4, ∆t = 0.05; ε = 1/4,h = 1/8, ∆t = 0.0125; which corresponds to meshing strategy h = O(ε), ∆t = O(ε2).

From Tables 3.1,2&3 and Fig. 3.1, we can draw the following observations:

Our numerical method for MD is of spectral order accuracy in space and second orderaccuracy in time, and conserves the density up to 12-digits. In the ‘nonrelativistic’ limitregime for initial data with O(ε)-wavelength, i.e. 0 < ε 1, the ε-resolution is: h = O(ε)and ∆t = O(ε2). Furthermore, our additional numerical experiments confirm that themethod is unconditionally stable, and show that meshing strategy: h = O(ε) and ∆t =O(ε) gives ‘incorrect’ numerical results in ‘nonrelativistic’ limit regime. However, for initialdata with O(1)-wavelength, much relaxed meshing strategy in the ‘nonrelativistic’ limitregime is admissible.

Bibliography

[1] S. Abenda, Solitary waves for Maxwell-Dirac and Coulomb-Dirac models, Ann. Inst.H. Poincare Phys. Theor. 68 (1998), pp. 229–244.

[2] A. Aftalion A, and Q. Du, Vortices in a rotating Bose-Einstein condensate: Criticalangular velocities and energy diagrams in the Thomas-Fermi regime, Phys. Rev. A,64(2001), pp. 063603.

[3] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell,Science 269, 198 (1995).

[4] W. Bao, Ground states and dynamics of multi-component Bose-Einstein condensates,SIAM Multiscale Modeling and Simulation, 2 (2004), pp. 210-236.

[5] W. Bao, Numerical methods for nonlinear Schrodinger equation under nonzero far-field conditions, preprint.

[6] W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein conden-sates by a normalized gradient flow, SIAM J. Sci. Comp., 25(2004), pp. 1674-1697.

[7] W. Bao, D. Jaksch, An explicit unconditionally stable numerical methods for solv-ing damped nonlinear Schrodinger equations with a focusing nonlinearity, SIAM J.Numer. Anal., 41(2003), pp. 1406-1426.

[8] W. Bao, D. Jaksch and P.A. Markowich, Numerical solution of the Gross-Pitaevskiiequation for Bose-Einstein condensation, J. Comput. Phys., 187(2003), pp. 318-342.

[9] W. Bao, D. Jaksch and P.A. Markowich, Three dimensional simulation of jet for-mation in collapsing condensates, J. Phys. B: At. Mol. Opt. Phys., 37(2004), pp.329-343.

[10] W. Bao, S. Jin and P.A. Markowich, On time-splitting spectral approximations forthe Schrodinger equation in the semiclassical regime, J. Comput. Phys., 175(2002),pp. 487-524.

[11] W. Bao, S. Jin and P.A. Markowich, Numerical study of time-splitting spectraldiscretizations of nonlinear Schrodinger equations in the semi-clasical regimes, SIAMJ. Sci. Comp., 25(2003), pp. 27-64.

[12] W. Bao and X.-G. Li, An efficient and stable numerical method for the Maxwell-Dirac system, J. Comput. Phys., 199 (2004), pp. 663-687.

99

100 BIBLIOGRAPHY

[13] W. Bao and J. Shen, A fourth-order time-splitting Laguerre-Hermite pseudo-spectralmethod for Bose-Einstein condensates, SIAM J. Sci. Comput., to appear.

[14] W. Bao, F. Sun, Efficient and stable numerical methods for the generalized andvector Zakharov system, SIAM J. Sci. Comput., to appear.

[15] W. Bao, F.F. Sun and G.W. Wei, Numerical methods for the generalized Zakharovsystem, J. Comput. Phys. 190(2003), pp. 201 - 228.

[16] W. Bao and W. Tang, Ground state solution of trapped interacting Bose-Einsteincondensate by directly minimizing the energy functional, J. Comput. Phys.,187(2003), pp. 230-254.

[17] W. Bao and Y. Zhang, Dynamics of the ground state and central vortex states inBose-Einstein condensation, Math. Mod. Meth. Appl. Sci., to appear.

[18] W. Bao, H.Q. Wang and P.A. Markowich, Ground, symmetric and central vortexstates in rotating Bose-Einstein condensates, Comm. Math. Sci., to appear.

[19] G. Baym and C. J. Pethick, Phys. Rev. Lett. 76, 6 (1996).

[20] P. Bechouche, N. J. Mauser and F. Poupaud, (Semi)-nonrelativistic limits of theDirac equation with external time-dependent electromagnetic field, Commun. Math.Phys., 197 (1998), pp. 405-425.

[21] J. Bolte and S. Keppeler, A semiclassical approach to the Dirac equation, Ann.Phys., 274 (1999), pp. 125-162.

[22] H. S. Booth, G. Legg and P. D. Jarvis, Algebraic solution for the vector potential inthe Dirac equation, J. Phys. A: Math. Gen., 34 (2001), pp. 5667-5677.

[23] J. Bourgain, On the Cauchy and invariant measure problem for the periodic Zakharovsystem, Duke Math. J. 76(1994), pp. 175-202.

[24] J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat.Math. Res. Notices 11(1996), pp. 515-546.

[25] N. Bournaveas, Local existence for the Maxwell-Dirac equations in three space di-mensions, Comm. Partial Differential Equations 21 (1996), pp. 693–720.

[26] C.C. Bradley, C.A. Sackett, R.G. Hulet, Bose-Einstein condensation of lithium: Ob-servation of limited condensate number, Phys. Rev. Lett. 78 (1997), pp. 985-989.

[27] B. M. Caradoc-Davis, R. J. Ballagh, K. Burnett, Coherent dynamics of vortex for-mation in trapped Bose-Einstein condensates, Phys. Rev. Lett. 83 (1999) 895.

[28] Q. Chang and H. Jiang, A conservative difference scheme for the Zakharov equations,J. Comput. Phys. 113(1994), pp. 309-319.

[29] J. M. Chadam, Global solutions of the Cauchy problem for the (classical) coupledMaxwell-Dirac equations in one space dimension, J. Funct. Anal. 13 (1973), pp.173-184.

BIBLIOGRAPHY 101

[30] T.F. Chan, D. Lee and L. Shen, Stable explicit schemes for equations of theSchrodinger type, SIAM J. Numer. Anal. 23, 274-281, 1986.

[31] T.F. Chan and L. Shen, Stability analysis of difference scheme for variable coefficientSchrodinger type equations, SIAM J. Numer. Anal. 24, 336-349, 1987.

[32] Q. Chang, B. Guo and H. Jiang, Finite difference method for generalized Zakharovequations, Math. Comp. 64(1995), pp. 537-553.

[33] J. Colliander, Wellposedness for Zakharov systems with generalized nonlinearity, J.Diff. Eqs. 148(1998), pp. 351-363.

[34] F. Dalfovo S. Giorgini, L.P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463(1999).

[35] A. Das, General solutions of Maxwell-Dirac equations in 1+1-dimensional space-timeand a spatially confined solution, J. Math. Phys, 34 (1993), pp. 3986-3985.

[36] A. Das, An ongoing big bang model in the special relativistic Maxwell-Dirac equa-tions, J. Math. Phys., 37(1996), pp. 2253-2259.

[37] A. Das and D. Kay, A class of exact plane wave solutions of the Maxwell- Diracequations, J. Math. Phys., 30 (1989), pp. 2280-2284.

[38] K.B. Davis, M.O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn,and W. Ketterle, Phys. Rev. Lett 75, 3969 (1995).

[39] A.S. Davydov, Phys. Scr. 20(1979), pp. 387.

[40] L.M. Degtyarev, V.G. Nakhan’kov and L.I. Rudakov, Dynamics of the formation andinteraction of Langmuir solitons and strong turbulence, Sov. Phys. JETP 40(1974),pp. 264-268.

[41] B. Desjardins, C.K. Lin and T.C. Tso, Semiclassical limit of the derivative nonlinearSchrodinger equation, M3AS 10, 261-285, 2000.

[42] P.A.M. Dirac, Principles of Quantum Mechanics, Oxford University Press, London(1958).

[43] M. Edwards, P.A. Ruprecht, K. Burnett, R.J. Dodd, and C.W. Clark,Phys. Rev. Lett. 77, 1671 (1996); D.A.W. Hutchinson, E. Zaremba, and A. Grif-fin, Phys. Rev. Lett. 78, 1842 (1997).

[44] M. Edwards and K. Burnett, Numerical solution of the nonlinear Schrodinger equa-tion for small samples of trapped neutral atoms, Phys. Rev. A 51, 1382 (1995).

[45] M. Esteban, E. Sere, An overview on linear and nonlinear Dirac equations, DiscreteContin. Dyn. Syst. 8 (2002), pp. 381-397.

[46] C. Fermanian-Kammerer, Semi-classical analysis of a Dirac equaiton without adia-batic decoupling, Monatsh. Math., to appear.

102 BIBLIOGRAPHY

[47] A. L. Fetter and A. A. Svidzinsky, Vortices in a trapped dilute Bose-Einstein con-densate, J. Phys.: Condens. Matter 13 (2001), pp. 135-194.

[48] G. Fibich, An adiabatic law for self-focusing of optical beams, Opt. Lett. 21 (1996),1735-1737.

[49] G. Fibich, Small beam nonparaxiality arrests self-focusing of optical beams, Phys.Rev. Lett. 76 (1996), 4356-4359.

[50] G. Fibich, V. Malkin and G. Papanicolaou, Beam self-focusing in the presence ofsmall normal time dispersion, Phys. Rev. A 52 (1995), 4218-4228.

[51] M. Flato, J. C. H. Simon and E. Taflin, Asymptotic completeness, global existenceand the infrared problem for the Maxwell-Dirac equations, Mem. Amer. Math. Soc.127 (1997), pp. 311.

[52] E. Forest and R.D. Ruth, Fourth order symplectic integration, Phys. D 43, 105(1990).

[53] B. Fornberg and T.A. Driscoll, A fast spectral algorithm for nonlinear wave equationswith linear dispersion, J. Comput. Phys. 155, 456 (1999).

[54] J. J. Garcia-Ripoll, V. M. Perez-Garcia and V. Vekslerchik, Construction of exactsolutions by spatial translations in inhomogeneous nonlinear Schrodinger equaitons,Phys. Rev. E 64(2001), 056602.

[55] I. Gasser and P.A. Markowich, Quantum hydrodynamics, Wigner transforms andthe classical limit, Asymptotic Analysis 14, 97-116, 1997.

[56] P. Gerard, Microlocal defect measures, Comm. PDE. 16, 1761-1794, 1991.

[57] V. Georgiev, Small amplitude solutions of the Maxwell-Dirac equations, IndianaUniv. Math. J. 40 (1991), pp. 845-883.

[58] P. Gerard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limitsand Wigner transforms, Comm. Pure Appl. Math. 50 (1997), pp. 321-377.

[59] J. Ginibre, Y. Tsutsumi and G. Velo, The Cauchy problem for the Zakharov system,J. Funt. Anal. 151(1997), pp. 384-436.

[60] R. Glassey, On the asymptotic behavior of nonlinear Schrodinger equations, Trans.Amer. Math. Soc. 182 (1973), 187-200.

[61] R. Glassey, On the blowing-up of solutions to the Cauchy problem for nonlinearSchrodinger equations, J. Math. Phys. 18 (1977), 1794-1797.

[62] R. Glassey, Approximate solutions to the Zakharov equations via finite differences,J. Comput. Phys. 100(1992), pp. 377-383.

[63] R. Glassey, Convergence of an energy-preserving scheme for the Zakharov equationsin one space dimension, Math. Comp. 58(1992), pp. 83-102.

BIBLIOGRAPHY 103

[64] O. Goubet and I. Moise, Attractor for dissipative Zakharov system, Nonlinear Anal.,Theory, Methods & Applications 31(1998), pp. 823-847.

[65] D. Gottlieb and S.A. Orszag, Numerical analysis of spectral methods (Soc. for In-dustr. & Appl. Math., Philadelphia, 1977).

[66] E. Grenier, Semiclassical limit of the nonlinear Schrodinger equation in small time,Proc. Amer. Math. Soc., 126(1998), pp. 523-530.

[67] M. Greiner, O. Mandel, T. Esslinger, T.W. Hansch, and I. Bloch, Quantum phasetransition from a superfluid to a mott insulator in a gas of ultracold atoms, Nature,415(2002), pp. 39-45.

[68] A. Griffin, D. Snoke, S. Stringaro (Eds.), Bose-Einstein condensation, CambridgeUniversity Press, New York, 1995.

[69] E.P. Gross, Nuovo. Cimento., 20(1961), pp. 454.

[70] L. Gross, The Cauchy problem for coupled Maxwell and Dirac equaitons, Comm.Pure Appl. Math. 19 (1966), pp. 1-15.

[71] D.S. Hall, M.R. Mattthews, J.R. Ensher, C.E. Wieman and E.A. Cornell, Dynamicsof component separation in a binary mixture of Bose-Einstein condensates, Phys.Rev. Lett., 81 (1998), pp. 1539-1542.

[72] H. Hadouaj, B. A. Malomed and G.A. Maugin, Soliton-soliton collisions in a gener-alized Zakharov system, Phys. Rev. A 44(1991), pp. 3932-3940.

[73] H. Hadouaj, B. A. Malomed and G.A. Maugin, Dynamics of a soliton in a generalizedZakharov system with dissipation, Phys. Rev. A 44(1991), pp. 3925-3931.

[74] R.H. Hardin and F.D. Tappert, Applications of the split-step Fourier method to thenumerical solution of nonlinear and variable coefficient wave equations, SIAM Rev.Chronicle 15, 423 (1973).

[75] Z.Y. Huang, S. Jin, P.A. Markowich, C. Sparber and C.X. Zheng, A time-splittingspectral scheme for the Maxwell-Dirac system, preprint.

[76] W. Hunziker, On the nonrelativistic limit of the Dirac theory, Commun. Math. Phys.40 (1975), pp. 215-222.

[77] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Cold bosonic atomsin optical lattices, Phys. Rev. Lett., 81(1998), pp. 3108-3111.

[78] S. Jin, P.A. Markowich and C.X. Zheng, Numerical simulation of a generalized Za-kharov system, J. Comput. Phys., to appear.

[79] A. Jungel, Nonlinear problems in quantum semiconductor modeling, Nonlin. Anal.,Proceedings of WCNA2000, to appear.

[80] A. Jungel, Quasi-hydrodynamic semiconductor equations, Progress in Nonlinear Dif-ferential Equations and Its Applications, Birkhauser, Basel, 2001.

104 BIBLIOGRAPHY

[81] T. Kato, On nonlinear Schrodinger equations, Ann. Inst. H. Poincare. Phus. Theor46 (1987), 113-129.

[82] M. Landman, G. Papanicolaou, C. Sulem and P. Sulem, Rate of blowup for solutionsof the nonlinear Schrodinger equation at critical dimension, Phys. Rev. A 38 (1988),3837-3843.

[83] M. Landman, G. Papanicolaou, C. Sulem, P. Sulem and X. Wang, Stability ofisotropic singularities for the nonlinear Schrodinger equation, Physica D 47 (1991),393-415.

[84] L. Laudau and E. Lifschitz, Quantum Mechanics: non-relativistic theory, PergamonPress, New York, 1977.

[85] P. Leboeuf and N. Pavloff, Phys. Rev. A 64, 033602 (2001); V. Dunjko, V. Lorent,and M. Olshanii, Phys. Rev. Lett. 86, 5413 (2001).

[86] A. J. Leggett, Bose-Einstein condensation in the alkali gases: some fundamentalconcepts, Rev. Mod. Phys. 73 (2001), pp. 307-356.

[87] E. H. Lieb, R. Seiringer, J. Yugvason, Bosons in a trap: a rigorous derivation of theGross-Pitaevskii energy functional, Phys. Rev. A 61, (2000) 3602.

[88] F. Lin and T. C. Lin, Vortices in two-dimensional Bose-Einstein condensates. Geom-etry and nonlinear partial differential equations (Hangzhou, 2001), 87–114, AMS/IPStud. Adv. Math., 29, Amer. Math. Soc., Providence, RI, 2002.

[89] M.J. Landman, G.C. Papanicolaou, C. Sulem, P.L. Sulem, X.P. Wang, Stability ofisotropic singularities for the nonlinear Schrodinger equation, Phys. D, 47 (1991),pp. 393–415.

[90] A. G. Lisi, A solitary wave solution of the Maxwell-Dirac equations, J. Phys. A:Math. Gen., 28 (1995), pp. 5385-5392.

[91] P.A. Markowich, N.J. Mauser and F. Poupaud, A Wigner function approach tosemiclassical limits: electrons in a periodic potential, J. Math. Phys. 35, 1066-1094,1994.

[92] P.A. Markowich, P. Pietra and C. Pohl, Numerical approximation of quadratic ob-servables of Schrodinger-type equations in the semi-classical limit, Numer. Math. 81,595-630, 1999.

[93] N. Masmoudi and N. J. Mauser, The selfconsistent Pauli equaiton, Monatsh. Math.,132 (2001), pp. 19–24.

[94] R. McLachlan, Symplectic integration of Hamiltonian wave equations, Numer. Math.66, 465 (1994).

[95] V. Masselin, A result of the blow-up rate for the Zakharov system in dimension 3,SIAM J. Math. Anal. 33 (2001), pp. 440-447.

BIBLIOGRAPHY 105

[96] B. Najman, The nonrelativstic limit of the nonlinear Dirac equaiton, Ann. Inst.Henri Poincare Non Lineaire 9 (1992), pp. 3-12.

[97] P.K. Newton, Wave interactions in the singular Zakharov system, J. Math. Phys.32(1991), pp. 431-440.

[98] G.C. Papanicolaou, C. Sulem, P.L. Sulem, X.P. Wang, Singular solutions of theZakharov equations for Langmuir turbulence, Phys. Fluids B, 3 (1991), pp. 969-980.

[99] A.S. Parkins and D.F. Walls, Physics Reports 303, 1 (1998).

[100] G.L. Payne, D.R. Nicholson, and R.M. Downie, Numerical solution of the Zakharovequations, J. Comput. Phys. 50(1983), pp. 482-498.

[101] N.R. Pereira, Collisions between Langmuir solitons, The Physics of Fluids 20(1977),pp. 750-755.

[102] C.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, CambridgeUniversity Press, 2002.

[103] L.P. Pitaevskii, Zh. Eksp. Teor. Fiz. 40, 646, 1961. (Sov. Phys. JETP 13, 451, 1961).

[104] L. Pitaevskii and S. Stringari, Bose-Einstein condensation, Oxford University Press,Oxford, 2002.

[105] D. S. Rokhsar, Phys. Rev. Lett. 79, 2164 (1997); R. Dum, J.I. Cirac, M. Lewenstein,and P. Zoller, Phys. Rev. Lett. 80, 2972 (1998); P. O. Fedichev, and G. V. Shlyap-nikov, Phys. Rev. A 60, R1779 (1999).

[106] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applicationsto geometric problems, Annals of Math., 118(1983), pp. 525-571.

[107] E.I. Schulman, Dokl. Akad. Nauk. SSSR 259, 579 [Sov. Phys. Dokl. 26, 691 (1981)].

[108] C. Sparber and P. Markowich, Semiclassical asymptotics for the Maxwell-Dirac sys-tem, J. Math. Phys. 44 (2003), pp. 4555–4572.

[109] H. Spohn, Semiclassical limit of the Dirac equaiton and spin precession, Ann. Phys.282 (2000), pp. 420-431.

[110] L. Stenflo, Phys. Scr. 33(1986), pp. 156.

[111] G. Strang, On the construction and comparison of differential schemes, SIAM J.Numer. Anal. 5, 506 (1968).

[112] C. Sulem and P.L. Sulem, Regularity properties for the equations of Langmuir tur-bulence, C. R. Acad. Sci. Paris Ser. A Math. 289(1979), pp. 173-176.

[113] C. Sulem and P.L. Sulem, The nonlinear Schrodinger equation, Springer, 1999.

[114] F.F. Sun, Numerical studies on the Zakharov system, Master thesis, National Uni-versity of Singapore, 2003.

106 BIBLIOGRAPHY

[115] T.R. Taha and M.J. Ablowitz, Analytical and numerical aspects of certain nonlin-ear evolution equations, II. Numerical, nonlinear Schrodinger equation, J. Comput.Phys. 55, 203 (1984).

[116] B. Thaller, The Dirac equation, New York, Springer, 1992.

[117] M. Weinstein, Nonlinear Schrodinger equations and sharp interpolation estimates,Comm. Math. Phys. 87 (1983), 567-576.

[118] M. Weinstein, Modulational stability of ground states of nonlinear Schrodinger equa-tions, SIAM J. Math. Anal. 16 (1985), 472-490.

[119] M. Weinstein, The nonlinear Schrodinger equations-singularity formation, stabilityand dispersion, Contemporary mathematics 99 (1989), 213-232.

[120] H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A.,150(1990), pp. 262-268.

[121] V.E. Zakharov, Zh. Eksp. Teor. Fiz. 62, 1745 (1972) [Sov. Phys. JETP 35, 908(1972)].

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