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Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

The nonlinear Schrödinger equation on star

graphs

Schrödinger and other dynamics on thin networks

Claudio Cacciapuoti

Hausdorff Center for MathematicsBonn Universität

Workshop“Mathematical Aspects of Quantum Mechanics and Quantum Transport Theory”

Bielefeld, April 23 - 28, 2012

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

1. Metric graphs: introduction & linear operators

2. Nonlinear Schrödinger equation on graphs(joint work with R. Adami, D. Finco, D. Noja)

- Scattering of fast solitons

- Stationary states

3. Approximation of networks of thin tubes(joint work with S. Albeverio, D. Finco)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Metric Graphs

A metric graph is realized by a set ofedges {ej}nj=1 and vertices, with a met-ric structure on any edge.Self-adjoint, (one-dimensional) differ-ential operators can be defined on thegraph.

Graphs as one-dimensional approx-imations for constrained dynamicsin which transverse dimensions aresmall with respect to longitudinalones.

Graphs are somewhat simplified objects which neglect much of the relevant(i.e. geometric) structure, yet they exhibit non trivial features.

- G. Berkolaiko et al., Quantum graphs and their applications, 2006.

- P. Exner et al., Analysis on graphs and its applications, 2008.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Metric Graphs

A metric graph is realized by a set ofedges {ej}nj=1 and vertices, with a met-ric structure on any edge.Self-adjoint, (one-dimensional) differ-ential operators can be defined on thegraph.

Graphs as one-dimensional approx-imations for constrained dynamicsin which transverse dimensions aresmall with respect to longitudinalones.

Graphs are somewhat simplified objects which neglect much of the relevant(i.e. geometric) structure, yet they exhibit non trivial features.

- G. Berkolaiko et al., Quantum graphs and their applications, 2006.

- P. Exner et al., Analysis on graphs and its applications, 2008.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Sobolev Spaces on graphs

Every edge of the graph is isomorphicto a (bounded or unbounded) orientedsegment, ej ∼ Ij

A function on a graph is a vector

Ψ = (ψ1, ..., ψN ) with ψj ≡ ψj (xj ) ; xj ∈ Ij

The spaces Lp(G) are defined by

Lp(G) =N⊕

j=1

Lp(Ij ) ; ‖Ψ‖p =

(N∑

j=1

‖ψj‖pLp (Ij )

) 1p

In a similar way one can define Hp(G) =⊕N

j=1 Hp(Ij )

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Laplacian on a star graph

For a Star Graph: ej ∼ (0,+∞), v ≡ 0

Hilbert Space

H =N⊕

j=1

L2((0,∞))

Ψ = (ψ1, . . . , ψN ) ∈ H

‖Ψ‖ =( N∑

j=1

‖ψj‖2L2(R+))1/2

The operator −∆G

−∆GΨ =(−d

2ψ1dx21

, . . . ,−d2ψNdx2N

)

D(−∆G) =N⊕

j=1

H2((0,∞)) + self-adjoint conditions in the vertex

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Laplacian on a star graph

For a Star Graph: ej ∼ (0,+∞), v ≡ 0Hilbert Space

H =N⊕

j=1

L2((0,∞))

Ψ = (ψ1, . . . , ψN ) ∈ H

‖Ψ‖ =( N∑

j=1

‖ψj‖2L2(R+))1/2

The operator −∆G

−∆GΨ =(−d

2ψ1dx21

, . . . ,−d2ψNdx2N

)

D(−∆G) =N⊕

j=1

H2((0,∞)) + self-adjoint conditions in the vertex

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Laplacian on a star graph

For a Star Graph: ej ∼ (0,+∞), v ≡ 0Hilbert Space

H =N⊕

j=1

L2((0,∞))

Ψ = (ψ1, . . . , ψN ) ∈ H

‖Ψ‖ =( N∑

j=1

‖ψj‖2L2(R+))1/2

The operator −∆G

−∆GΨ =(−d

2ψ1dx21

, . . . ,−d2ψNdx2N

)

D(−∆G) =N⊕

j=1

H2((0,∞)) + self-adjoint conditions in the vertex

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Laplacian on a star graph

For a Star Graph: ej ∼ (0,+∞), v ≡ 0Hilbert Space

H =N⊕

j=1

L2((0,∞))

Ψ = (ψ1, . . . , ψN ) ∈ H

‖Ψ‖ =( N∑

j=1

‖ψj‖2L2(R+))1/2

The operator −∆G

−∆GΨ =(−d

2ψ1dx21

, . . . ,−d2ψNdx2N

)

D(−∆G) =N⊕

j=1

H2((0,∞)) + self-adjoint conditions in the vertex

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Laplacian on a Star Graph: Vertex Conditions

Any unitary N × N matrix, U, defines a selfadjoint Laplacian on G, −∆UG , bythe boundary conditions

(U − 1)

ψ1(v)...ψN (v)

+ i(U + 1)ψ

′1(v)...

ψ′N (v)

= 0

- Dirichlet condition [decoupling condition]:

ψ1(v) = ψ2(v) = · · · = ψN (v) = 0

- Kirchhoff condition [standard condition]:

ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑

j=1

ψ′j (v) = 0

- delta-like condition: [Ujk = 2(N + iα)−1 − δjk ]

ψ(v) ≡ ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑

j=1

ψ′j (v) = αψ(v) α ∈ R

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Laplacian on a Star Graph: Vertex Conditions

Any unitary N × N matrix, U, defines a selfadjoint Laplacian on G, −∆UG , bythe boundary conditions

(U − 1)

ψ1(v)...ψN (v)

+ i(U + 1)ψ

′1(v)...

ψ′N (v)

= 0- Dirichlet condition [decoupling condition]:

ψ1(v) = ψ2(v) = · · · = ψN (v) = 0

- Kirchhoff condition [standard condition]:

ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑

j=1

ψ′j (v) = 0

- delta-like condition: [Ujk = 2(N + iα)−1 − δjk ]

ψ(v) ≡ ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑

j=1

ψ′j (v) = αψ(v) α ∈ R

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Laplacian on a Star Graph: Vertex Conditions

Any unitary N × N matrix, U, defines a selfadjoint Laplacian on G, −∆UG , bythe boundary conditions

(U − 1)

ψ1(v)...ψN (v)

+ i(U + 1)ψ

′1(v)...

ψ′N (v)

= 0- Dirichlet condition [decoupling condition]:

ψ1(v) = ψ2(v) = · · · = ψN (v) = 0

- Kirchhoff condition [standard condition]:

ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑

j=1

ψ′j (v) = 0

- delta-like condition: [Ujk = 2(N + iα)−1 − δjk ]

ψ(v) ≡ ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑

j=1

ψ′j (v) = αψ(v) α ∈ R

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Laplacian on a Star Graph: Vertex Conditions

Any unitary N × N matrix, U, defines a selfadjoint Laplacian on G, −∆UG , bythe boundary conditions

(U − 1)

ψ1(v)...ψN (v)

+ i(U + 1)ψ

′1(v)...

ψ′N (v)

= 0- Dirichlet condition [decoupling condition]:

ψ1(v) = ψ2(v) = · · · = ψN (v) = 0

- Kirchhoff condition [standard condition]:

ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑

j=1

ψ′j (v) = 0

- delta-like condition: [Ujk = 2(N + iα)−1 − δjk ]

ψ(v) ≡ ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑

j=1

ψ′j (v) = αψ(v) α ∈ R

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Nonlinear Schrödinger Equation on a Star Graph (focusing, withpower nonlinearity)

Let µ > 0

id

dtΨt = HΨt − |Ψt |2µΨt |Ψ|2µ ≡ diag [|ψj |2µ]

Or in integral form

Ψt = e−iHtΨ0 + i

∫ t0

e−iH(t−s)|Ψs |2µΨs ds

Or in components

i∂

∂tψj (xj , t) = −

∂2

∂x2jψj (xj , t)− |ψj (xj , t)|2µψj (xj , t) xj > 0

but again do not forget the boundary conditions.

Nature of the problem: a system of N PDEs coupled through the vertexconditions.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Nonlinear Schrödinger Equation on a Star Graph (focusing, withpower nonlinearity)

Let µ > 0

id

dtΨt = HΨt − |Ψt |2µΨt |Ψ|2µ ≡ diag [|ψj |2µ]

Or in integral form

Ψt = e−iHtΨ0 + i

∫ t0

e−iH(t−s)|Ψs |2µΨs ds

Or in components

i∂

∂tψj (xj , t) = −

∂2

∂x2jψj (xj , t)− |ψj (xj , t)|2µψj (xj , t) xj > 0

but again do not forget the boundary conditions.

Nature of the problem: a system of N PDEs coupled through the vertexconditions.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Nonlinear Schrödinger Equation on a Star Graph (focusing, withpower nonlinearity)

Let µ > 0

id

dtΨt = HΨt − |Ψt |2µΨt |Ψ|2µ ≡ diag [|ψj |2µ]

Or in integral form

Ψt = e−iHtΨ0 + i

∫ t0

e−iH(t−s)|Ψs |2µΨs ds

Or in components

i∂

∂tψj (xj , t) = −

∂2

∂x2jψj (xj , t)− |ψj (xj , t)|2µψj (xj , t) xj > 0

but again do not forget the boundary conditions.

Nature of the problem: a system of N PDEs coupled through the vertexconditions.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Laplacian with delta-like condition in the vertex

From now on we shall consider only delta-like conditions in the vertex.

ψ(v) ≡ ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑

j=1

ψ′j (v) = αψ(v) α ∈ R

We denote by Hα the corresponding Laplacian

When α = 0 (Kirchhoff condition) we write H ≡ Hα=0

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Soliton on the line

Consider the equation

i∂

∂tψ(x , t) = − ∂

2

∂x2ψ(x , t)− |ψ(x , t)|2µψ(x , t) x ∈ R , t > 0

Given the functionφ(x) = [(µ+ 1)]

12µ sech

1µ (µx)

One has the family of solitary rotating/traveling waves:

φx0,v,ω(x , t) := ei v

2xe−it

v2

4 e iωtω1

2µ φ(√ω(x − x0 − vt))

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Are there solitons on Start Graphs?

Yes, for star graphs with even number of edges and Kirchhoff conditions in thevertex.Think at the star graph as N

2lines which cross at the vertex

On each line put a copy of the soliton φx0,v,ω(x , t) with the same x0, v and ωThe function Φx0,v,ω constructed in this way satisfies the Kirchhoff conditionin the vertexΦx0,v,ω describes

N2

identical solitons moving simultaneously on the graph

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Are there solitons on Start Graphs?

Yes, for star graphs with even number of edges and Kirchhoff conditions in thevertex.Think at the star graph as N

2lines which cross at the vertex

On each line put a copy of the soliton φx0,v,ω(x , t) with the same x0, v and ωThe function Φx0,v,ω constructed in this way satisfies the Kirchhoff conditionin the vertexΦx0,v,ω describes

N2

identical solitons moving simultaneously on the graph

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks

Is this picture (approximately) realistic?

t = 0 t large

Inspired by the work: J. Holmer, J. Marzuola, and M. Zworski, Fast solitonscattering by delta impurities, Commun. Math. Phys. 274, 2007.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks

Is this picture (approximately) realistic?

t = 0 t large

Setting:

Cubic NLS

Kirchhoff vertex (but more general conditions are allowed)

Initial stateΨ0(x) = (

√2χ(x)e−i

v2

x sech(x − x0), 0, 0)

χ is a cutoff function

High velocity regime v � 1x0 > v1−δ, with 0 < δ < 1.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks

Find an approximate solution of the equation:

Ψt = e−iHtΨ0 + i

∫ t0

e−iH(t−s)|Ψs |2Ψs ds

Three step analysis

Phase 1: approaching the vertex

Phase 2: scattering through the vertex

Phase 3: persistence of the outgoing state

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 1, approaching the vertex

t ∈ [0, t1 = x0/v − v−δ]

In this phase the incoming pulse moves from x0 to x0 − vt1 = v 1−δ,remaining at a distance of order v 1−δ from the vertex [0 < δ < 1]

During this phase only a small tail of the pulse intersects the vertex, thesolution Ψt behaves as the solitary solution of the NLS in R and remainssubstantially supported only on the edge e1

The approximating function is:

Φt(x) = (φx0,−v (x , t), 0, 0)

Lemma

For any t ∈ [0, t1]‖Ψt − Φt‖ 6 Ce−v

1−δ

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 1, approaching the vertex

t ∈ [0, t1 = x0/v − v−δ]

In this phase the incoming pulse moves from x0 to x0 − vt1 = v 1−δ,remaining at a distance of order v 1−δ from the vertex [0 < δ < 1]

During this phase only a small tail of the pulse intersects the vertex, thesolution Ψt behaves as the solitary solution of the NLS in R and remainssubstantially supported only on the edge e1

The approximating function is:

Φt(x) = (φx0,−v (x , t), 0, 0)

Lemma

For any t ∈ [0, t1]‖Ψt − Φt‖ 6 Ce−v

1−δ

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 1, approaching the vertex

t ∈ [0, t1 = x0/v − v−δ]

In this phase the incoming pulse moves from x0 to x0 − vt1 = v 1−δ,remaining at a distance of order v 1−δ from the vertex [0 < δ < 1]

During this phase only a small tail of the pulse intersects the vertex, thesolution Ψt behaves as the solitary solution of the NLS in R and remainssubstantially supported only on the edge e1

The approximating function is:

Φt(x) = (φx0,−v (x , t), 0, 0)

Lemma

For any t ∈ [0, t1]‖Ψt − Φt‖ 6 Ce−v

1−δ

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 2, scattering through thevertex

t ∈ [t1 = x0/v − v−δ, t2 = x0/v + v−δ]

During this phase the “body” of the soliton crosses the vertex.

The time interval t2 − t1 is small (of order v−δ, 0 < δ < 1). Then theeffect of the nonlinear term is negligible

The pulse travels for a large distance (of order v(t2 − t1) = v 1−δ,0 < δ < 1). Then the linear dynamics can be described by using ascattering approximation.

Let T and R be the transmission and reflection coefficients of the linearoperator

T =2

NR = −N − 2

N

Recall that the function Ψ(k)

Ψ(k, x1, x2, x3) = (e−ikx1 + Re ikx1 ,T e ikx2 ,T e ikx3 )

satisfies the Kirchhoff boundary conditions, and is a solution ofHΨ = k2Ψ, in distributional sense.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 2, scattering through thevertex

t ∈ [t1 = x0/v − v−δ, t2 = x0/v + v−δ]

During this phase the “body” of the soliton crosses the vertex.

The time interval t2 − t1 is small (of order v−δ, 0 < δ < 1). Then theeffect of the nonlinear term is negligible

The pulse travels for a large distance (of order v(t2 − t1) = v 1−δ,0 < δ < 1). Then the linear dynamics can be described by using ascattering approximation.

Let T and R be the transmission and reflection coefficients of the linearoperator

T =2

NR = −N − 2

N

Recall that the function Ψ(k)

Ψ(k, x1, x2, x3) = (e−ikx1 + Re ikx1 ,T e ikx2 ,T e ikx3 )

satisfies the Kirchhoff boundary conditions, and is a solution ofHΨ = k2Ψ, in distributional sense.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 2, scattering through thevertex

t ∈ [t1 = x0/v − v−δ, t2 = x0/v + v−δ]

Approximating function:

ΦSt =

(φx0,−v (t) + R φ−x0,v (t),T φ−x0,v (t),T φ−x0,v (t)

)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 2, scattering through thevertex

t ∈ [t1 = x0/v − v−δ, t2 = x0/v + v−δ]

Approximating function:

ΦSt =

(φx0,−v (t) + R φ−x0,v (t),T φ−x0,v (t),T φ−x0,v (t)

)

On the left side the state ΦS at time t1. On the right side the state ΦS at time t2.

Each edge of the graph is extended to a line by ideally adding the half line (−∞, 0]represented by the dashed lines. Dashed lines are not part of the real graph.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 2, scattering through thevertex

t ∈ [t1 = x0/v − v−δ, t2 = x0/v + v−δ]

Approximating function:

ΦSt =

(φx0,−v (t) + R φ−x0,v (t),T φ−x0,v (t),T φ−x0,v (t)

)

Lemma

For any t ∈ [t1, t2]‖Ψt − ΦSt ‖ 6 Cv−δ/2

for v large enough and some constant δ ∈ (0, 1).

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 3, persistence of theoutgoing state

t ∈ [t2 = x0/v + v−δ, t3 = t2 + T ln v ]

Phase 3 begins with three low solitons which move away from the vertex

For t > t2 and x ∈ R we define the functions φtr and φref byi∂

∂tφtr = − ∂

2

∂x2φtr − |φtr |2φtr

φtr (x , t2) = T φ−x0,v (x , t2)

i∂

∂tφref = − ∂

2

∂x2φref − |φref |2φref

φref (x , t2) = R φ−x0,v (x , t2)

The outgoing state is approximated by

Φoutt :=(φref (t), φtr (t), φtr (t)

)

Lemma

Fix T > 0, then for any time t ∈ [t2, t2 + T ln v ], there exists 0 < η < 1/2such that

‖Ψt − Φoutt ‖ 6 Cv−η

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 3, persistence of theoutgoing state

t ∈ [t2 = x0/v + v−δ, t3 = t2 + T ln v ]

Phase 3 begins with three low solitons which move away from the vertexFor t > t2 and x ∈ R we define the functions φtr and φref by

i∂

∂tφtr = − ∂

2

∂x2φtr − |φtr |2φtr

φtr (x , t2) = T φ−x0,v (x , t2)

i∂

∂tφref = − ∂

2

∂x2φref − |φref |2φref

φref (x , t2) = R φ−x0,v (x , t2)

The outgoing state is approximated by

Φoutt :=(φref (t), φtr (t), φtr (t)

)

Lemma

Fix T > 0, then for any time t ∈ [t2, t2 + T ln v ], there exists 0 < η < 1/2such that

‖Ψt − Φoutt ‖ 6 Cv−η

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 3, persistence of theoutgoing state

t ∈ [t2 = x0/v + v−δ, t3 = t2 + T ln v ]

Phase 3 begins with three low solitons which move away from the vertexFor t > t2 and x ∈ R we define the functions φtr and φref by

i∂

∂tφtr = − ∂

2

∂x2φtr − |φtr |2φtr

φtr (x , t2) = T φ−x0,v (x , t2)

i∂

∂tφref = − ∂

2

∂x2φref − |φref |2φref

φref (x , t2) = R φ−x0,v (x , t2)

The outgoing state is approximated by

Φoutt :=(φref (t), φtr (t), φtr (t)

)

Lemma

Fix T > 0, then for any time t ∈ [t2, t2 + T ln v ], there exists 0 < η < 1/2such that

‖Ψt − Φoutt ‖ 6 Cv−η

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 3, persistence of theoutgoing state

t ∈ [t2 = x0/v + v−δ, t3 = t2 + T ln v ]

Phase 3 begins with three low solitons which move away from the vertexFor t > t2 and x ∈ R we define the functions φtr and φref by

i∂

∂tφtr = − ∂

2

∂x2φtr − |φtr |2φtr

φtr (x , t2) = T φ−x0,v (x , t2)

i∂

∂tφref = − ∂

2

∂x2φref − |φref |2φref

φref (x , t2) = R φ−x0,v (x , t2)

The outgoing state is approximated by

Φoutt :=(φref (t), φtr (t), φtr (t)

)

Lemma

Fix T > 0, then for any time t ∈ [t2, t2 + T ln v ], there exists 0 < η < 1/2such that

‖Ψt − Φoutt ‖ 6 Cv−η

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 3, persistence of theoutgoing state

Proposition (HMZ)

For 0 < |γ| < 1, let φγt be defined byi∂

∂tφγ = − ∂

2

∂x2φγ − |φγ |2φγ

φγ(x , 0) = γ√

2 cosh−1(x)

x ∈ R

then the function φγt has the following asymptotic behavior for t →∞

φγ(x , t) =

{a√

2 cosh−1(ax) e iϕ(γ) +OL∞(t−1/2) 1/2 < |γ| < 1

OL∞(t−1/2) 0 < |γ| < 1/2

with a = 2|γ| − 1.

For N = 3, T = 2/3 and |R| = 1/2. The transmitted low solitons survive,the reflected low soliton disappears.For N ≥ 4, T 6 1/2 and |R| > 1/2. The transmitted low solitons disappear,only the reflected low soliton survives.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 3, persistence of theoutgoing state

Final outcome of the scattering

N = 3 N ≥ 4

Warning:

Our result holds up to times of order ln v , with an error OL2 (v−η). Theresult in Proposition [HMZ] approximates φtr (t) and φref (t) for t →∞with an error OL∞(t−1/2).Our estimates are in L2-norm. The result in Proposition [HMZ] is inL∞-norm. Our result is rigorous for what concerns “mass transmission”not for the “profile” of the outcoming pulses

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Propagation of pulses in networks: Phase 3, persistence of theoutgoing state

Final outcome of the scattering

N = 3 N ≥ 4Warning:

Our result holds up to times of order ln v , with an error OL2 (v−η). Theresult in Proposition [HMZ] approximates φtr (t) and φref (t) for t →∞with an error OL∞(t−1/2).Our estimates are in L2-norm. The result in Proposition [HMZ] is inL∞-norm. Our result is rigorous for what concerns “mass transmission”not for the “profile” of the outcoming pulses

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

References

W. K. Abu Salem, J. Fröhlich, and I. M. Sigal, Colliding solitons for the

nonlinear Schrödinger equation, Comm. Math. Phys. 291 (2009), 151–176.

R. Adami, C. C., D. Finco, and D. Noja, Fast solitons on star graphs, Rev.

Math. Phys 23 (2011), 409–451.

K. Datchev and J. Holmer, Fast soliton scattering by attractive delta impurities,

Comm. Part. Diff. Eq. 34 (2009), 1074–1113.

J. Holmer, J. Marzuola, and M. Zworski, Fast soliton scattering by delta

impurities, Commun. Math. Phys. 274 (2007), 187–216.

P. G. Kevrekidis, D. J. Frantzeskakis, G. Theocharis, and I. G. Kevrekidis,

Guidance of matter waves through Y-junctions, Phys. Lett. A 317 (2003),513–522.

G. Perelman, A remark on soliton-potential interaction for nonlinear Schrödinger

equations, Math. Res. Lett. 16 (2009), no. 3, 477–486.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Standing waves

Consider the focusing NLS equation with a δ vertex of strength α

id

dtΨt = HαΨt − |Ψt |2µΨt

where Hα denotes the Laplacian on the graph with a delta-like condition inthe vertex

We want to study possible existence and properties of stationary solutions(standing waves):

Ψt = eiωt Φω

The amplitude Φω satisfies the “elliptic” equation

HαΦω − |Φω|2µΦω = −ωΦω

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Standing waves

Now on every edge

−φ′′ − |φ|2µφ = −ωφ φ ∈ L2(R+)

and the most general solution is

φ(σ, a; x) = σ [(µ+ 1)ω]1

2µ sech1µ (µ√ω(x − a)) , |σ| = 1 a ∈ R

so that(Φω)i = φ(σi , ai ) ,

where σi , ai have to be chosen to satisfy Φω ∈ D(Hα)

The continuity at the vertex implies σj = 1

aj = εja, εi = ±1 , a ≥ 0

For a > 0 there are “bumps” and “tails”- εj = 1: there is a “bump” on the edge j- εj = −1: there is a “tail” on the edge j

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Standing waves

The condition on the derivative

n∑j=1

(Φω)′j (0) = α (Φω)1 (0)

gives

tanh(µ√ωa)

N∑i=1

εi =α√ω

(1)

Consequence:∑Ni=1 εi must have the same sign of α

- α > 0 strictly more bumps than tails- α < 0 strictly more tails than bumps- α = 0 same number of tails and bumps or a = 0

For every configuration of ε1, ..., εN (up to permutations of the edges) thecondition (1) fixes uniquely a

We index the stationary states with the number j of bumps

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Standing waves: α 6= 0 (delta-like vertex)

Stationary solutions exist and (their amplitudes) are given by

(Φjω

)i

=

{φ(aj ) i = 1, . . . , j

φ(−aj ) i = j + 1, . . . ,N

aj =1

µ√ω

arctanh

(α

(2j − N)√ω

)

Due to the constraint between number of bumps and the sign of α

α > 0 : Φjω with j = [N/2 + 1], . . . ,N

α < 0 : Φjω with j = 0, . . . , [(N − 1)/2]

For any value of α 6= 0 there are[N + 1

2

]states

Finally, there is a lower bound on the allowed frequencies:

α2

N2< ω

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Standing waves: α 6= 0 (delta-like vertex)

Nonlinear stationary states:α < 0 , N = 3 , j = 0, 1

Nonlinear stationary states:

α > 0 , N = 3 , j = 2, 3

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Standing waves: α = 0 (Kirchhoff vertex)

Let us consider the Kirchhoff case, α = 0.From the boundary condition

tanh(µ√ωa)

N∑i=1

εi = 0 (2)

- First case (N odd): (2) =⇒ a = 0The stationary state is unique

(Φω)i (x) = φ(0, x) i = 1, . . . ,N

N half solitons continuously joint at the vertex

- Second case (N even): (2) =⇒ a ∈ R,∑N

i=1 εi = 0There is a one-parameter family of stationary states

(Φaω)i (x) =

{φ(−a, x) i = 1, . . .N/2φ(+a, x) i = N/2 + 1, . . .N

a ∈ R

N/2 identical solitons

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Standing waves: α = 0 (Kirchhoff vertex)

N odd

N even

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Variational properties

Minimize the energy at constant mass

inf{E [Ψ] s.t. Ψ ∈ E ,M[Ψ] ≡ ||Ψ||2 = m} (3)

Recall that

D(E) = {Ψ ∈ H1(G) s.t. ψ1(0) = ... = ψN (0)}

and

E(Ψ) =1

2‖Ψ′‖2 + α|ψ1(0)|2 −

1

2µ+ 2‖Ψt‖2µ+22µ+2

Theorem (Work in progress)

Let µ > 0, α < α∗ < 0, ω > α2

N2. Then the minimum problem (3) attains a

solution coinciding with the N tail state Φ0ω

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Variational properties

A technical difficulty: use of symmetric decreasing rearrangements Ψ∗ of Ψ toreduce the minimization problem to the set of symmetric states and prove theexistence of the minimum.

Proposition (Modified Polya-Szego inequality)

Assume that Ψ ∈ D(E). Then ‖Ψ∗‖Lp (G) = ‖Ψ‖Lp (G), for any 1 6 p 6∞, and‖Ψ∗′‖2 6 N

2

4‖Ψ′‖2

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Variational properties

A simple example.

Same L∞ and L1-norms, slopes grow to have ‖Ψ∗′‖2 = N2

4‖Ψ′‖2.

‖Ψ′‖2 = 2 ; ‖Ψ∗′‖2 = 92

so that

‖Ψ∗′‖2 = N2

4‖Ψ′‖2 (N = 3)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Variational properties: energy minima (N=3, Cubic NLS, Kirchhoffvertex)

It turns out that for cubic nonlinearity, α = 0 and N = 3 the energy does notattain a minimum value on the set of states with fixed mass

The stationary state is indeed a saddle point

Take Ψ ∈ D(E) such that ‖Ψ‖2 = m

Assume ‖ψ1‖2L2(R+) 6 ‖ψ2‖2L2(R+) 6 ‖ψ3‖

2L2(R+) and set

m1 = ‖ψ1‖2L2(R+) ; m2 = ‖ψ2‖2L2(R+) + ‖ψ3‖

2L2(R+)

Define

Φxm1,m2 (x1, x2, x3) :=

m1√

2sech

(m12x1)

m22√

2sech

(m24

(x2 − x))

m22√

2sech

(m24

(x3 + x))

where 0 < 2m1 6 m2, x ≥ 0, and the following condition of “continuity at thevertex” holds:

m1 =m22

sech(m2

4x)

moreoverm1 + m2 = m

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Variational properties: energy minima (N=3, Cubic NLS, Kirchhoffvertex)

ThenE(Ψ) ≥ E(Φxm1,m2 )

and E is minimized on states of the form Φxm1,m2Moreover E(Φxm1,m2 ) is increasing in m1 and limm1→0 E(Φ

xm1,m2 ) = −

m3

96then

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Variational properties: energy minima (N=3, Cubic NLS, Kirchhoffvertex)

Theorem

For any Ψ ∈ D(E) such that ‖Ψ‖2 = m, the following inequality holds

E(Ψ) > −m3

96.

The infimum cannot be reached, as for m1 = 0 the conditions

m1 =m22

sech(m2

4x)

m1 + m2 = m

do not correspond to an admissible state.

The state Φxm1,m2 with x = 0, m1 =m3

and m2 =2m3

is a minimum of theenergy on the manifold of states with Ψ1 = Ψ2 = Ψ3, and it is a maximum onthe manifold of states of the form Φxm1,m2 . Then it is a saddle point.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Summary on NLS on graphs

- Summary

Analysis of the scattering of fast solitons

Existence of stationary states

Proof of the nonexistence of the energy minimum (Kirchhoff graph)

- Work in progress

Minimization of the energy functional for fixed mass(concentration-compactess)

Existence of the ground state and characterization as minimizer of theaction functional

Stability of the ground state (Grillakis-Shatah-Strauss)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

References

R. Adami, C. C., D. Finco, and D. Noja, arXiv:1202.2890 [math-ph] (to appear

on J.Phys. A).

R. Adami, C. C., D. Finco, and D. Noja, arXiv:1104.3839 [math-ph].

T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes inMathematics, AMS, vol 10, Providence, 2003.

Fukuizumi R., Ohta M., Ozawa T.: Ann. I.H.Poincaré, AN, 25, 837-845 (2008)

Grillakis M., Shatah J., Strauss W., J.Funct.Anal., 74, 160-197 (1987)

Vakhitov M.G., Kolokolov A.A., Radiophys. Quantum Electron 16, 783 (1973)

Weinstein M., Comm.Pure Appl.Math. 39, 51-68 (1986)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Vertex coupling approximation

Any unitary N × N matrix, U, defines a selfadjoint Laplacian on G, −∆UG , bythe boundary conditions

(U − 1)

ψ1(v)...ψN (v)

+ i(U + 1)ψ

′1(v)...

ψ′N (v)

= 0Problem: Which unitary matrices U define “good” conditions in the vertex?

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Vertex coupling approximation

Consider a manifold Ω surrounding the graph G

Define −∆Ω (boundary conditions on ∂Ω)Analyze the convergence of −∆Ω to −∆UG as Ω collapses onto G(convergence of the spectrum, resolvent convergence, convergence of thesolution of a Cauchy problem, ...)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Vertex coupling approximation

Consider a manifold Ω surrounding the graph GDefine −∆Ω (boundary conditions on ∂Ω)

Analyze the convergence of −∆Ω to −∆UG as Ω collapses onto G(convergence of the spectrum, resolvent convergence, convergence of thesolution of a Cauchy problem, ...)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Vertex coupling approximation

Consider a manifold Ω surrounding the graph GDefine −∆Ω (boundary conditions on ∂Ω)Analyze the convergence of −∆Ω to −∆UG as Ω collapses onto G(convergence of the spectrum, resolvent convergence, convergence of thesolution of a Cauchy problem, ...)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Vertex coupling approximation

Neumann boundary conditions on ∂Ω:M. I. Freidlin and A. D. Wentzel ’93; G. Raugel ’95; S. Kosugi ’00;P. Kuchment and H. Zeng ’01; J. Rubinstein and M. Schatzman ’01; Y. Saitō’01; P. Exner and O. Post ’05; O. Post ’06; A. I. Bonciocat ’08.

The essential spectrum of −∆Ω is [0,∞]

Let ψ be a function in D(−∆Ω) with finite energy, ‖ψ‖H1(Ω) 6 C then:

in the edges, far away from the vertex region, ψ is essentially constant inthe transverse direction

in vertex region ψ is essentially constant

In the limit one gets Kirchhoff conditions in the vertex

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Vertex coupling approximation

Dirichlet boundary conditions on ∂Ω:O. Post ’05; S. Molchanov and B. Vainberg ’06; D. Grieser ’07; S. Albeverio,C. C., and D. Finco ’07; C. C. and P. Exner ’07; C. C. and D. Finco ’08;G. Dell’Antonio and E. Costa ’10.

The essential spectrum of −∆Ω is[π2

d2,+∞

)where d is the diameter of

the tubes, d → 0

A rescaling is needed −∆Ω → −∆Ω − π2

d2

The operator −∆Ω− π2

d2can have eigenvalues which go to −∞ as d → 0

No intuition for the shape of the function in the vertex region

The conditions in the vertex depend on the properties of theapproximating manifold

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Waveguide Collapsing onto a Graph

=⇒

Dirichlet Laplacian on Ω: D(−∆DΩ ) :={ψ ∈ H2(Ω) s.t. ψ

∣∣∂Ω

= 0}

The Waveguide Ω

C : R→ R2 C(s) := {(γ1(s), γ2(s))| s ∈ R} ; γ′1(s)2 + γ′2(s)2 = 1

C has no self-intersectionsSigned curvature: γ(s) := γ′2(s)γ

′′1 (s)− γ′1(s)γ′′2 (s) ; γ ∈ C∞0 (R)

Ω :={

(x , y) ∈ R2∣∣∣ (x , y) = (γ1(s), γ2(s)) + un̂(s), ∀s ∈ R, u ∈ (−1, 1)}

n̂(s) = (−γ′2, γ′1) is the vector (of unit norm) orthogonal to CΩ is a waveguide of constant width (Supp |γ| < 1)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Waveguide Collapsing onto a Graph

Scaling of Ω: ε is a “small” dimensionless parameter

Global System of Coordinates: s ∈ R, u ∈ (0, 1)

Scaling of the Width: u → εau a > 3Scaling of the Curvature:

γ(s)→ γε(s) :=1

εγ( sε

); θ =

∫Rγ(s)ds =

∫Rγε(s)ds = θε

Scaling of the Waveguide: Ω→ ΩεΩε is net of waveguides which collapses onto a “prototypical” graph asε→ 0Two length scales

A small one εa (a > 3), Transversal “Fast Dynamics”A large one ε, Longitudinal “Slow Dynamics”

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Waveguide Collapsing onto a Graph

Scaling of Ω: ε is a “small” dimensionless parameter

Global System of Coordinates: s ∈ R, u ∈ (0, 1)Scaling of the Width: u → εau a > 3

Scaling of the Curvature:

γ(s)→ γε(s) :=1

εγ( sε

); θ =

∫Rγ(s)ds =

∫Rγε(s)ds = θε

Scaling of the Waveguide: Ω→ ΩεΩε is net of waveguides which collapses onto a “prototypical” graph asε→ 0Two length scales

A small one εa (a > 3), Transversal “Fast Dynamics”A large one ε, Longitudinal “Slow Dynamics”

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Waveguide Collapsing onto a Graph

Scaling of Ω: ε is a “small” dimensionless parameter

Global System of Coordinates: s ∈ R, u ∈ (0, 1)Scaling of the Width: u → εau a > 3Scaling of the Curvature:

γ(s)→ γε(s) :=1

εγ( sε

); θ =

∫Rγ(s)ds =

∫Rγε(s)ds = θε

Scaling of the Waveguide: Ω→ ΩεΩε is net of waveguides which collapses onto a “prototypical” graph asε→ 0Two length scales

A small one εa (a > 3), Transversal “Fast Dynamics”A large one ε, Longitudinal “Slow Dynamics”

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Waveguide Collapsing onto a Graph

Scaling of Ω: ε is a “small” dimensionless parameter

Global System of Coordinates: s ∈ R, u ∈ (0, 1)Scaling of the Width: u → εau a > 3Scaling of the Curvature:

γ(s)→ γε(s) :=1

εγ( sε

); θ =

∫Rγ(s)ds =

∫Rγε(s)ds = θε

Scaling of the Waveguide: Ω→ Ωε

Ωε is net of waveguides which collapses onto a “prototypical” graph asε→ 0Two length scales

A small one εa (a > 3), Transversal “Fast Dynamics”A large one ε, Longitudinal “Slow Dynamics”

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Waveguide Collapsing onto a Graph

Scaling of Ω: ε is a “small” dimensionless parameter

Global System of Coordinates: s ∈ R, u ∈ (0, 1)Scaling of the Width: u → εau a > 3Scaling of the Curvature:

γ(s)→ γε(s) :=1

εγ( sε

); θ =

∫Rγ(s)ds =

∫Rγε(s)ds = θε

Scaling of the Waveguide: Ω→ ΩεΩε is net of waveguides which collapses onto a “prototypical” graph asε→ 0

Two length scalesA small one εa (a > 3), Transversal “Fast Dynamics”A large one ε, Longitudinal “Slow Dynamics”

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Waveguide Collapsing onto a Graph

Scaling of Ω: ε is a “small” dimensionless parameter

γ(s)→ γε(s) :=1

εγ( sε

); θ =

∫Rγ(s)ds =

∫Rγε(s)ds = θε

A small one εa (a > 3), Transversal “Fast Dynamics”A large one ε, Longitudinal “Slow Dynamics”

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Reduction to a One Dimensional Dynamics

The operator −∆DΩε is unitarily equivalent to Hε in L2(R× (0, 1))

Hε = −∂

∂s

1

(1 + εa−1uγ(s/ε))2∂

∂s− 1ε2a

∂2

∂u2+

1

ε2Wε(s, u) ,

Wε(s, u) = −γ(s/ε)2

4+O(εa−1)

ψ̃ ∈ D(Hε) ⊂ L2(R× (0, 1)) =⇒ ψ̃∣∣

u=0,1= 0

Dimensional reduction(φ0,

(Hε −

π2

ε2a

)φ0

)L2((0,1))

'(− d

2

ds2− 1ε2γ2(s/ε)

4

)with

φ0(u) =

√2

εasin(πu/εa)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Reduction to a One Dimensional Dynamics

The operator −∆DΩε is unitarily equivalent to Hε in L2(R× (0, 1))

Hε = −∂

∂s

1

(1 + εa−1uγ(s/ε))2∂

∂s− 1ε2a

∂2

∂u2+

1

ε2Wε(s, u) ,

Wε(s, u) = −γ(s/ε)2

4+O(εa−1)

ψ̃ ∈ D(Hε) ⊂ L2(R× (0, 1)) =⇒ ψ̃∣∣

u=0,1= 0

Dimensional reduction(φ0,

(Hε −

π2

ε2a

)φ0

)L2((0,1))

'(− d

2

ds2− 1ε2γ2(s/ε)

4

)with

φ0(u) =

√2

εasin(πu/εa)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Analysis of the One Dimensional Problem

Definition (Zero energy resonance)

Assume that ec|·|v ∈ L1(R) for some c > 0. We say that the Hamiltonian

h = − d2

ds2+ v(s)

has a zero energy resonance if there exists ψr ∈ L∞(R), ψr /∈ L2(R) such thathψr = 0 in distributional sense

If h has a zero energy resonance we define

ρ1 = limx→−∞

ψr (x) and ρ2 = limx→+∞

ψr (x)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Main Theorem

Theorem (Albeverio, C.C., Finco ’07)

Assume that γ ∈ C∞0 (R) and take a > 3, then two cases can occur:

1. H = − d2

ds2− γ2/4 has no zero energy resonances(

φ0,

(Hε −

π2

ε2a

)φ0

)L2((0,1))

ε→0−−−→ −∆DG

−∆DG f := (−f ′′1 ,−f ′′2 ) ; f1(0) = f2(0) = 0

2. H = − d2

ds2− γ2/4 has a zero energy resonance ψr(

φ0,

(Hε −

π2

ε2a

)φ0

)L2((0,1))

ε→0−−−→ −∆ρ1ρ2G

−∆ρ1ρ2G f := (−f′′

1 ,−f ′′2 ) ;ρ2f1(0) = ρ1f2(0)

ρ1f′

1 (0) + ρ2f′

2 (0) = 0

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Conclusions

Dirichlet case (Albeverio, C.C., Finco ’07)the result does not change if the projection is taken on excited transversestates

Generic case: no zero energy resonance, decoupling conditionsNon-generic case: existence of a zero energy resonance, coupling betweenthe edgesSmall perturbations of the geometry (C.C., Exner ’07)

γε(s) :=

√1 + λε

εγ( sε

)λ ∈ R

θε =

(1 +

λ

2ε

)θ +O(ε2)

In the non generic case the parameter λ enters into the definition of theboundary conditions of the operator on the graph

ρ2f1(0) = ρ1f2(0)

ρ1f′

1 (0) + ρ2f′

2 (0) =λ̂

2

(ρ1f1(0) + ρ2f2(0)

)λ̂ := −λ

∫ ∞−∞

γ(s)2

4|ψr (s)|ds

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Conclusions

Dirichlet case (Albeverio, C.C., Finco ’07)the result does not change if the projection is taken on excited transversestatesGeneric case: no zero energy resonance, decoupling conditionsNon-generic case: existence of a zero energy resonance, coupling betweenthe edges

Small perturbations of the geometry (C.C., Exner ’07)

γε(s) :=

√1 + λε

εγ( sε

)λ ∈ R

θε =

(1 +

λ

2ε

)θ +O(ε2)

In the non generic case the parameter λ enters into the definition of theboundary conditions of the operator on the graph

ρ2f1(0) = ρ1f2(0)

ρ1f′

1 (0) + ρ2f′

2 (0) =λ̂

2

(ρ1f1(0) + ρ2f2(0)

)λ̂ := −λ

∫ ∞−∞

γ(s)2

4|ψr (s)|ds

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Conclusions

Dirichlet case (Albeverio, C.C., Finco ’07)the result does not change if the projection is taken on excited transversestatesGeneric case: no zero energy resonance, decoupling conditionsNon-generic case: existence of a zero energy resonance, coupling betweenthe edgesSmall perturbations of the geometry (C.C., Exner ’07)

γε(s) :=

√1 + λε

εγ( sε

)λ ∈ R

θε =

(1 +

λ

2ε

)θ +O(ε2)

In the non generic case the parameter λ enters into the definition of theboundary conditions of the operator on the graph

ρ2f1(0) = ρ1f2(0)

ρ1f′

1 (0) + ρ2f′

2 (0) =λ̂

2

(ρ1f1(0) + ρ2f2(0)

)λ̂ := −λ

∫ ∞−∞

γ(s)2

4|ψr (s)|ds

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Conclusions

Neumann case (C.C., Finco ’10):

ψ ∈ D(−∆NΩ) , ∂nψ|∂Ω = 0

Reduction of the dynamics with respect to the ground transverse state:standard condition in the vertex

(φN0 ,HNε φ

N0 )L2((−1,1)) ' −

d2

ds2

Reduction of the dynamics with respect to excited transverse states:decoupling conditions in the vertex (no possibility of zero energyresonances!)

(φNn ,HNε φ

Nn )L2((−1,1)) '

(−

d2

ds2+

3

4

1

ε2γ2(s/ε) +

E Nnε2a

)n > 1

Robin case (C.C., Finco ’10): ψ ∈ D(−∆RΩ), ∂nψ|∂Ω +αψ|∂Ω = 0, α ∈ RDependence on the transverse energyExistence of a generic and non-generic case

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Conclusions

Neumann case (C.C., Finco ’10):

ψ ∈ D(−∆NΩ) , ∂nψ|∂Ω = 0

Reduction of the dynamics with respect to the ground transverse state:standard condition in the vertex

(φN0 ,HNε φ

N0 )L2((−1,1)) ' −

d2

ds2

Reduction of the dynamics with respect to excited transverse states:decoupling conditions in the vertex (no possibility of zero energyresonances!)

(φNn ,HNε φ

Nn )L2((−1,1)) '

(−

d2

ds2+

3

4

1

ε2γ2(s/ε) +

E Nnε2a

)n > 1

Robin case (C.C., Finco ’10): ψ ∈ D(−∆RΩ), ∂nψ|∂Ω +αψ|∂Ω = 0, α ∈ RDependence on the transverse energyExistence of a generic and non-generic case

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Conclusions

Neumann case (C.C., Finco ’10):

ψ ∈ D(−∆NΩ) , ∂nψ|∂Ω = 0

Reduction of the dynamics with respect to the ground transverse state:standard condition in the vertex

(φN0 ,HNε φ

N0 )L2((−1,1)) ' −

d2

ds2

Reduction of the dynamics with respect to excited transverse states:decoupling conditions in the vertex (no possibility of zero energyresonances!)

(φNn ,HNε φ

Nn )L2((−1,1)) '

(−

d2

ds2+

3

4

1

ε2γ2(s/ε) +

E Nnε2a

)n > 1

Robin case (C.C., Finco ’10): ψ ∈ D(−∆RΩ), ∂nψ|∂Ω +αψ|∂Ω = 0, α ∈ RDependence on the transverse energyExistence of a generic and non-generic case

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

References

Dirichlet boundary:

O. Post, Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case, J. Phys.

A: Math. Gen. 38 (2005), no. 22, 4917–4931.

G. Dell’Antonio and L. Tenuta, Quantum graphs as holonomic constraints, J. Math. Phys. 47 (2006),

072102.

S. Albeverio, C. C., and D. Finco, Coupling in the singular limit of thin quantum waveguides, J. Math.

Phys. 48 (2007), 032103.

C. C. and P. Exner, Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent

waveguide, J. Phys. A: Math. Theor. 40 (2007), no. 26, F511–F523.

C. C. and D. Finco, Graph-like models for thin waveguides with Robin boundary conditions,

arxiv:0803.4314 [math-ph] (2008), 27pp, accepted for publication in Asymptotic Analysis.

S. Molchanov and B. Vainberg, Laplace operator in networks of thin fibers: spectrum near the threshold,

Stochastic analysis in mathematical physics, Proceedings of a Satellite Conference of ICM 2006 (2006),69–93, edited by G. Ben Arous, A.-B. Cruzeiro, Y. Le Jan, J.-C. Zambrini.

S. Molchanov and B. Vainberg, Transition from a network of thin fibers to the quantum graph: an

explicitly solvable model, Contemp. Math., AMS 415 (2006), 227–239.

S. Molchanov and B. Vainberg, Scattering solutions in a network of thin fibers: small diameter

asymptotics, Commun. Math. Phys. 273 (2007), 533–559.

D. Grieser, Spectra of graph neighborhoods and scattering, Proc. London Math. Soc. 97 (2008), no. 3,

718–752.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

References

Dirichlet boundary (continues):

M. Harmer, B. Pavlov, and A. Yafyasov, Boundary conditions at the junction, J. Comput. Electron. 6

(2007), 153–157.

G. Dell’Antonio and E. Costa, Effective Schroedinger dynamics on ε-thin Dirichlet waveguides via

Quantum Graphs I: star-shaped graphs, arXiv:1004.4750 [math-ph] (2010), 23pp.

S. Albeverio and S. Kusuoka, Diffusion processes in thin tubes and their limits on graphs, (2010).

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

References

Neumann boundary:

M. I. Freidlin and A. D. Wentzel, Diffusion processes on graphs and averaging principle, Ann. Probab. 21

(1993), no. 4, 2215–2245.

S. Kosugi, A semilinear elliptic equation in a thin network-shaped domain, J. Math. Soc. Japan 52 (2000),

no. 3, 673–697.

Y. Saitō, The limiting equation for Neumann Laplacians on shrinking domains, Electron. J. Diff. Equations

2000 (2000), no. 31, 1–25.

Y. Saitō, Convergence of the Neumann Laplacian on shrinking domains, Analysis (Munich) 21 (2001),

no. 2, 171–204.

J. Rubinstein and M. Schatzman, Variational problems on multiply connected thin strips. I. Basic estimates

and convergence of the Laplacian spectrum, Arch. Ration. Mech. Anal. 160 (2001), no. 4, 271–308.

P. Kuchment and H. Zeng, Convergence of spectra of mesoscopic systems collapsing onto a graph, J.

Math. Anal. Appl. 258 (2001), no. 2, 671–700.

K. Kuwae and T. Shioya, Convergence of spectral structures: a functional analytic theory and its

applications to spectral geometry, Comm. Anal. Geom. 11 (2003), no. 4, 599–673.

G. Raugel, Dynamics of partial differential equations on thin domains, Dynamical systems, Lecture Notes

in Math. 1609 (1995), 208–315.

P. Exner and O. Post, Convergence of spectra of graph-like thin manifolds, J. Geom. Phys. 54 (2005),

77–115.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

References

Neumann boundary (continues):

O. Post, Spectral convergence of quasi-one-dimensional spaces, Ann. Henri Poincaré 7 (2006), 933–973.

P. Exner and O. Post, Convergence of resonances on thin branched quantum waveguides, J. Math. Phys.

48 (2007), 092104, 43pp.

A. I. Bonciocat, Curvature bounds and heat kernels: discrete versus continuous spaces, Ph.D. Thesis,

Universität Bonn, http://hss.ulb.uni-bonn.de:90/2008/1497/1497.htm, 2008.

P. Exner and O. Post, Approximation of quantum graph vertex couplings by scaled Schrödinger operators

on thin branched manifolds, J. Phys. A: Math. Theo. 42 (2009), 415305, 22pp.

http://hss.ulb.uni-bonn.de:90/2008/1497/1497.htm

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Variational properties

We want to identify the ground state of the system among the finite energystates

E ={

Ψ ∈ H1(G) s.t. ψ1(0) = · · · = ψN (0)}

Recall that

E(Ψ) =1

2‖Ψ′‖2 + α|ψ1(0)|2 −

1

2µ+ 2‖Ψt‖2µ+22µ+2

The stationary states of the NLS on graphs turn out to be critical points(S ′ω[Ψ] = 0 coincides with the stationarity equation) of the so called actionfunctional Sω

Sω[Ψ] = E [Ψ] +ω

2||Ψ||22 .

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Variational properties

The action Sω[Ψ] is unbounded from below in E , but stationary states arecontained in the manifold

{Ψ ∈ E s.t. Iω[Ψ] ≡ 〈S ′ω[Ψ],Ψ〉 = 0}

Ground states, if they exist, are solutions of the following constrainedminimization problem

inf {Sω[Ψ] s.t. Ψ ∈ E , Iω[Ψ] = 0} ≡ d(ω)

Theorem

Let µ > 0, α < α∗ < 0, ω > α2

N2. Then the minimum problem for Sω[Ψ]

constrained on the natural manifold Iω[Ψ] = 0 attains a solution coincidingwith the N tail state Φ0ω

In this sense we consider the N tail state the ground state

Other examples of minimization for NLS on the line with point interactionsyeld similar problems (Fukuizumi-Ohta-Ozawa 2008, Fukuizumi-Jeanjean2008, Adami-Noja 2011)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Variational properties

A tecnical difficulty: use of symmetric decreasing rearrangements Ψ∗ of Ψ toreduce to the halfline and calculate the exact value of the inf Sω[Ψ] in theKirchhoff case, needed in the proof of existence of ground state.

Theorem (Modified Polya-Szego inequality)

Assume that Ψ ∈ E . Then ‖Ψ∗‖Lp (G) = ‖Ψ‖Lp (G), for any 1 6 p 6∞, and‖Ψ∗‖2H1(G) 6

N2

4‖Ψ‖2H1(G)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Variational properties

A simple example.

Same L∞ and L1-norms, slopes grow to have ‖Ψ∗‖2H1(G) 6N2

4‖Ψ‖2H1(G).

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Variational properties: energy minima

A slightly different variational problem: Minimize the energy at constant mass

inf{E [Ψ] s.t. Ψ ∈ E ,M[Ψ] ≡ ||Ψ||22 = m}

Recall thatE = {Ψ ∈ H1(G) s.t. ψ1(0) = ... = ψN (0)}

and

E(Ψ) =1

2‖Ψ′‖2 + α|ψ1(0)|2 −

1

2µ+ 2‖Ψt‖2µ+22µ+2

This second variational problem can be studied using the concentrationcompactness method of P. L. Lions (around 1982, problems in Rn).In the case of a star graphs there is no translation invariance, and the c.c.method must be adapted.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Variational properties: energy minima (N=3, Cubic NLS, Kirchhoffvertex)

It turns out that for cubic nonlinearity, α = 0 and N = 3 the energy does notattain a minimum value on the set of states with fixed mass

The stationary state is indeed a stationary point

Take Ψ ∈ E such that ‖Ψ‖2 = m

Assume ‖ψ1‖2L2(R+) 6 ‖ψ2‖2L2(R+) 6 ‖ψ3‖

2L2(R+) and set

m1 = ‖ψ1‖2L2(R+) ; m2 = ‖ψ2‖2L2(R+) + ‖ψ3‖

2L2(R+)

Define

Φxm1,m2 (x1, x2, x3) :=

m1√

2sech

(m12x1)

m22√

2sech

(m24

(x2 − x))

m22√

2sech

(m24

(x3 + x))

where 0 < m1 6 m2, x ≥ 0, and the following condition of “continuity at thevertex” holds:

m1 =m22

sech(m2

4x)

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Variational properties: energy minima (N=3, Cubic NLS, Kirchhoffvertex)

It turns out thatE(Ψ) ≥ E(Φxm1,m2 )

Then E attains its minimum on states of the form Φxm1,m2Moreover E(Φxm1,m2 ) is increasing in m1 and limm1→0 E(Φ

xm1,m2 ) = −

m3

96then

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Variational properties: energy minima (N=3, Cubic NLS, Kirchhoffvertex)

Theorem

For any Ψ ∈ H1(G) such that ‖Ψ‖2 = m, the following inequality holds

E(Ψ) > −m3

96.

The infimum cannot be achieved, as for m1 = 0 the condition

m1 =m22

sech(m2

4x)

does not correspond to an admissible state.

The state Φxm1,m2 with x = 0, m1 =m3

and m2 =2m3

is a minimum of theenergy on the manifold of states with Ψ1 = Ψ2 = Ψ3, and it is a maximum onthe manifold of states of the form Φxm1,m2 . Then it is a saddle point.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Orbital stability of the ground state

Due to the U(1) invariance of the dynamics, the stability has to be consideredas orbital stability.The orbit of Φω is defined as O(Φω) = {e iθΦω(x), θ ∈ R}.

The state Φω is orbitally stable if for every � > 0 there exists δ > 0 such that

d(Ψ(0),O(Φω)) < δ ⇒ d(Ψ(t),O(Φω)) < � ∀t > 0

whered(ψ,O(Φω)) = inf

u∈O(Φω)||ψ − u||E

and the norm || · ||E is the energy norm.

We shall use classical results on the analysis of orbital stability of solitarysolutions of nonlinear equations due to Weinstein, Grillakis-Shatah-Strauss(’80, KG, NLS, other equations).

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Orbital stability of the ground state

The NLS on a graph turns out to be a hamiltonian system on the real Hilbertspace of the couples of real and imaginary part of the wavefunction.Decomposing Ψ = u + iv , one obtains the canonical system

d

dt

(uv

)= JE ′[u, v ] , J =

(0 I−I 0

)where the Hamiltonian E coincides with the energy up to substitution of realvariables, E ≡ E [u, v ].Linearization of the hamiltonian system around the stationary state is achievedby posing

(Ψt)k = (Φ0ω,k + ηk + iρk )e

iωt

and neglecting higher order terms than linear. The η and ρ satisfy

d

dt

(ηρ

)= JL

(ηρ

).

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Orbital stability of the ground state

OperatorL = diag(L−,L+)

and

(L+)i,k =(− d

2

dx2+ ω − |Φ0ω,k |2µ

)δi,k

(L−)i,k =(− d

2

dx2+ ω − (2µ+ 1)|Φ0ω,k |2µ

)δi,k .

L− and L+ are matrix self adjoint operators acting on the real vectorfunctions η and ρ belonging to D(Hα).

The Weinstein and GSS theory implies that solitary solutions are orbitallystable if

i) spectral conditions hold:i1) kerL+ = {Φ0ω} and the rest of the spectrum is positive;i2) n(L−) = 1 where the left hand side is the number of negative eigenvalues.

ii) Vakhitov-Kolokolov condition ddω‖Φ0ω‖22 > 0 holds.

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Orbital stability of the ground state

Theorem

Let µ ∈ [0, 2], α < α∗ < 0, ω > α2

N2.

Then the ground state Φ0ω is orbitally stable in H1(G)

Note that from Vakhitov-Kolokolov condition, ddω‖Φ0ω‖22 > 0, it follows that

for µ > 2 there exists ω∗ such that Ψ0ω is orbitally stable for ω ∈ (α2

N2, ω∗)

and is orbitally unstable for ω > ω∗ .

Introduction

MetricGraphs

Laplacian(s)on graphs

NLS ongraphs

Propagationof pulses innetworks

Standingwaves

Squeezingof fatmanifolds

Conclusions

There exist nonlinear standing waves for the NLS on a star graphs

The standing waves are explicit in some relevant case (δ star graph,including Kirchhoff, and others not treated here)

The ground state is variationally characterized

It is (orbitally) stable in the case of attractive δ star graph

IntroductionMetric GraphsLaplacian(s) on graphsNLS on graphs

Propagation of pulses in networksStanding wavesSqueezing of fat manifolds