Stability in the energy space for chains of solitons of...

Stability in the energy spae for hains of solitons of the LandauLifshitz

equation

Andr de Laire (Universit Lille 1)

Joint work with Philippe Gravejat (ole polytehnique)

Shrdinger equations and appliations

CIRM June 18th 2014

Andr de Laire Stability for hains of solitons of the LL equation 1/1

Outline


The LandauLifshitz equation

The dynamis of magnetization in a ferromagneti material is given by the

LandauLifshitz equation.

The magnetization is a diretion eld:

~m(x , t) : RD R S2 R3, ~m = (m

1

,m2

,m3

)

|~m| = (m21

+m22

+m23

)1/2 = 1


Models

3D ferromagneti materials

2D: Thin materials.

1D: Cesium nikel triuoride CsNiF

3

, the oupling between the Ni ions is muh

stronger than the other ouplings in the lattie.



t ~m = ~m ~heff (~m)

precession

~heff (~m): eetive magneti eld



t ~m = ~m ~heff (~m)

precession

~m (~m ~heff (~m))

damping

~heff (~m): eetive magneti eld

> 0 : Gilbert damping oeient > 0, w.l.o.g. 2 + 2 = 1


The anisotropi undamped Landau-Lifshitz equation

~heff (~m) = ~m m3e3,

t ~m = ~m (~m m3e3),

~m(x , t) : RD R S2, ~m = (m

1

,m2

,m3

)



~heff (~m) = ~m m3e3,

t ~m = ~m (~m m3e3),

~m(x , t) : RD R S2, ~m = (m

1

,m2

,m3

)

tm1 = m2(m3 m3) +m3Sm2

tm2 = m3m1 m1(m3 + m3)

tm3 = m1m2 +m2m1



~heff (~m) = ~m m3e3,

t ~m = ~m (~m m3e3),

~m(x , t) : RD R S2, ~m = (m

1

,m2

,m3

)

The equation is hamiltonian. The onserved Hamiltonian is the Landau-Lifshitz energy

E(~m(t)) :=1

2

RN

|~m(x , t)|2 dx +

2

RN

m3

(x , t)2 dx = E(~m(0)), t R.

The anisotropy parameter 0:

= 0 : isotropi ase, Shrdinger map equation

> 0 : easy-plane or planar anisotropy lim|x|

m3

(x , t) = 0

In the sequel, we onsider = 1 and solutions m with nite Landau-Lifshitz energy.


The hydrodynami framework

When the map m := m1

+ im2

does not vanish, it may be written as

m =

1 m23

exp i.

The funtions v = m3

and w = are solutions to the hydrodynami Landau-Lifshitzequation

tv = div((1 v2)w

),

tw = (

v(|w |2 1) +v

1 v2+

v |v |2

(1 v2)2

)

.(HLL)

The linearized equation around the zero solution is dispersive with dispersion relation

2 = |k |4 + |k |2.

Dispersion veloity is at least equal to the sound speed cs = 1.


Solitons

We look for traveling waves solutions propagating along the x1

-axis with speed c, i.e.

~m(x) = ~u(x

1

ct, x2

, . . . , xD).

Then the prole

~u satises

c1

~u + ~u (~u u

3

e

3

) = 0.

Taking

~u and using that

~u (~u~u) = ~u(~u ~u)~u(~u ~u) = ~u|~u|2 ~u,

we obtain

~u = |~u|2~u+ u23

~u u

3

e

3

+ c~u 1

~u. (TWc)


Solitons

We look for traveling waves solutions propagating along the x1

-axis with speed c, i.e.

~m(x) = ~u(x

1

ct, x2

, . . . , xD).

Then the prole

~u satises

c1

~u + ~u (~u u

3

e

3

) = 0.

Taking

~u and using that

~u (~u~u) = ~u(~u ~u)~u(~u ~u) = ~u|~u|2 ~u,

we obtain

~u = |~u|2~u+ u23

~u u

3

e

3

+ c~u 1

~u. (TWc)

Trivial solutions: onstants in S1 {0}.

It is enough to onsider c 0.


Papaniolaou and Spathis (1999) investigated their existene and qualitative

properties for D = 2, 3. They derived numerially a branh of traveling waves forsubsoni speeds |c| < 1.

Lin and Wei (2010) proved the existene of small speed travelling waves with a

vortex-antivortex pair when D = 2.

d.L. (2014): the non-existene of non-onstant subsoni small energy travelling

waves for D = 2, 3, 4.

d.L. (2014): also smoothness and their algebrai deay at innity for D 2.


Some numerial results for D = 2 (PapaniolaouSpathis)

0 15 30 45 60

15

30

p

E

c 0

c 1

E = p

c 0.78



Funtion u3

for c = 0.5.



Figure (a): speed c = 0.2. Figure (b): speed c = 0.95.


From LandauLifshitz to Shrdinger

Let

~u be a solution of (TWc). Using the stereographi variable

=u1

+ iu2

1 + u3

,

we have that satises

+1 ||2

1+ ||2 + ic

1

=2

1 + ||2( ),

whih seems like a perturbed equation for the traveling waves for a Nonlinear Shrdinger

equation (GrossPitaevskii equation).


From LandauLifshitz to Shrdinger

Let

~u be a solution of (TWc). Using the stereographi variable

=u1

+ iu2

1 + u3

,

we have that satises

+1 ||2

1 + ||2 + ic

1

=2

1+ ||2( )

that seems like a perturbed equation for the traveling waves for a nonlinear Shrdinger

equation (GrossPitaevskii equation):

+ (1 ||2) ic1

= 0. (GP)


Solitons for D = 1

In dimension one, the traveling wave solutions are the following expliit dark solitons.

Lemma (The one-dimensional ase)

Let D = 1 and c 0. Assume that ~u is a nontrivial nite energy solution of (TWc).Then 0 c < 1 and (up to invarianes) the solution is given by

u1

= c seh(

1 c2 x), u2

= tanh(

1 c2 x), u3

=

1 c2 seh(

1 c2 x).


Multisolitons for D = 1

The Landau-Lifshitz equation is integrable by means of the inverse sattering method. In

partiular, there are expliit formulae for multi-solitons. They behave like a sum of

ordered solitons as t .

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05-08.aviMedia File (video/avi)

Multisolitons for D = 1

The Landau-Lifshitz equation is integrable by means of the inverse sattering method. In

partiular, there are expliit formulae for multi-solitons. They behave like a sum of

ordered solitons as t .

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02-05-08.aviMedia File (video/avi)

Chains of solitons

For c 6= 0, the soliton u =(1 u2

3

) 12

exp i is represented in the hydrodynami

framework by the pair

vc =(vc ,wc

)=

(u3

, x),

where

vc(x) =(1 c2)

1

2

osh((1 c2)1

2 x) ,

and

wc (x) =c vc (x)

1 vc(x)2.


Chains of solitons

For c 6= 0, the soliton u =(1 u2

3

) 12

exp i is represented in the hydrodynami

framework by the pair

vc =(vc ,wc

)=

(u3

, x),

where

vc(x) =(1 c2)

1

2

osh((1 c2)1

2 x) ,

and

wc (x) =c vc (x)

1 vc(x)2.

A hain of solitons is dened as a perturbation of a sum of solitons

Sc,a,s(x) =N

j=1

sj vcj (x aj) =N

j=1

(sj vcj (x aj ), sj wcj (x aj)

),

with a = (a1

, . . . , aN) RN, c = (c

1

, . . . , cN) (1, 1)Nand s = (s

1

, . . . , sN) {1}N.

The hain is well-prepared when the speeds are ordered a

ording to the positions, i.e.

c1

< < cN ,

for

a1

< < aN .


The Cauhy problem for the original Landau-Lifshitz equation

The energy spae is dened as

E(R) ={m : R S2, s.t. m L2(R) and m

3

L2(R)}.

Theorem (Zhou-Guo '84, Sulem-Sulem-Bardos '86)

Let m0 E(R). There exists a global solution m L(R, E(R)) to

tm +m (xxm m3e3) = 0, (LL)

with initial datum m0.

Remark: no uniqueness

Theorem (Existene and uniqueness of smooth solutions)

Let k 1 and m0 E(R), with [m0] Hk(R). There exists a unique global solutionm : R R S2 to (LL) with initial datum m0, suh that

tm L([T ,T ],Hk1(R)) and (m

3

, xm) L([T ,T ],Hk(R))2,

for any positive number T . The Landau-Lifshitz energy is onserved along the ow.

See Sulem-Sulem-Bardos '86, Chang-Shatah-Uhlenbek '00, MGahagan '04.


The Cauhy problem for the hydrodynami Landau-Lifshitz equation

In the hydrodynami framework, the Landau-Lifshitz energy is equal to

E(v ,w) =1

2

R

( (xv)2

1 v2+ (1 v2)w2 + v2

)

.

The non-vanishing spae is dened as

NV(R) ={v = (v ,w) H1(R) L2(R), s.t. max

xR|v(x)| < 1

}.

Another onserved quantity is the momentum:

P(v ,w) =

R

vw .

Theorem 1 (d.L-Gravejat, 2014)

Let v0 = (v0,w0) NV(R).

(i) There exists a number Tmax

> 0 and a unique solution v C0([0,Tmax

),NV(R)) to(HLL) with initial datum v

0

suh that there exist smooth solutions vn C(R [0,T ])

to (HLL) satisfying

vn v in C0([0,T ],NV(R)), as n , for any T < T

max

.

(ii) Tmax

is haraterized by limtTmax

maxxR |v(x , t)| = 1 if Tmax

Minimizing haraterization of solitons

For c 6= 0, the soliton vc is a minimizer of the variational problem

Emin

(pc ) = inf{E(v), v : R C s.t. P(v) = pc

}.

The speed c is related to the momentum pc through the formula

pc = 2 artan( (1 c)

1

2

c

)

.

The Euler-Lagrange equation is y

E(vc) = cP

(vc).

p

E

E = p

2


Orbital stability of well-prepared hains of solitons

Theorem 2 (d.L.-Gravejat 2014)

Let c0 (1, 1)N , s0 {1}N , and v0 = (v0,w0) NV(R). Assume that

c0

1

< < 0 < < c0N .

There exist four numbers > 0, > 0, A > 0 and L > 0, suh that, if

v

0 Sc0,a0,s0

H1L2

= 0 < ,

for positions a0 RN suh that

min

{a0

k+1 a0

k , 1 k N 1}= L0 > L,

then there exist a unique solution v C0(R+,NV(R)) to (HLL) with initial datum v0

, as

well as N funtions ak C1(R+,R), with ak(0) = a

0

k , suh that

a

k(t) c

0

k

A

(0 + e

L0),

and

v(, t) S

c0,a(t),s0

H1L2

A(0 + e

L0),

for any t R+.


Orbital stability of a single soliton

Corollary 3 (d.L-Gravejat 2014)

Let c0 (1, 0) (0, 1), a0 R, s0 {1}, and v0 NV(R). There exist two numbers > 0 and A > 0 suh that, if

v

0 s0 vc0( a0)

H1L2

< ,

then there exist a unique solution v C0(R+,NV(R)) to (HLL) with initial datum v0

, as

well as a funtion a C1(R+,R), with a(0) = a0

, suh that

a

(t) c0 A,

and

v(, t) s0 v

c0( a(t))

H1L2

A,

for any t R+.


Constrution of smooth solutions to (HLL)

It follows from the existene of smooth solutions to (LL).

Lemma

Let k 4 and set

NVk (R) ={v = (v ,w) Hk+1(R) Hk (R), s.t. max

xR|v(x)| < 1

}.

Given a pair v0 NVk(R), there exists a number T

max

> 0 and a unique solution

v L([0,Tmax

),NVk (R)) to (HLL) with initial datum v0. The maximal time Tmax

is

haraterized by the ondition

lim

tTmax

max

xR|v(x , t)| = 1, if T

max

< +.


Continuity of the ow map

Given a smooth solution v = (v ,w) to (HLL), we dene the map

=1

2

(xv

(1 v2)1

2

+ i(1 v2)1

2w)

exp i,

where the phase is given by

(x , t) =

x

v(y , t)w(y , t) dy .

We also set

F (v ,)(x , t) =

x

v(y , t)(y , t) dy .

The map is solution to the nonlinear Shrdinger equation

it+ xx+2||2+

1

2

v2

Re((1 2F (v ,))

)(1 2F (v ,)) = 0,

while the funtion v satises the system{

tv = 2x Im((2F (v ,) 1)

),

xv = 2Re((1 2F (v ,))

).

(Generalized Hasimoto: Chang, Shatah and Uhlenbek 2000)


The Strihartz estimates for the linear Shrdinger equation provide the ontinuity of the

ow map orresponding to this system.

Lemma

Let T > 0. Given two smooth solutions (v1

,1

) and (v2

,2

) to the previous systemwith initial datum (v0j ,

0

j ), there exist a number > 0, depending only on v0

j L2 and0j L2 , and a universal onstant K suh that

v

1

v2

C0([0,T ],L2)

+

1

2

C0([0,T ],L2)

K(v

0

1

v02

L2

+0

1

02

L2

),

for any T [0,min{,T}].

Theorem 1 derives from expressing this ontinuity property in terms of the pair

v = (v ,w).


Orbital stability of a single soliton

The strategy is similar to the one developed by Martel-Merle-Tsai '02, '05, for the

Korteweg-de Vries and the nonlinear Shrdinger equations (see also

Bthuel-Gravejat-Smets '12).

It results from the following quantiation of the minimizing nature of a soliton.

Lemma

Let c (1, 0) (0, 1), and set Hc = E(vc) cP

(vc). There exists a number c > 0suh that

Hc(), L2 c2

H1L2 ,

for any pair H1(R) L2(R) suh that

xvc , L2 = P(vc), L2 = 0.

When = v vc satises the two orthogonality onditions, we have

E(v) cP(v)= E(vc) cP(vc) +1

2

Hc(), L2 +O(3H1L2

)

E(vc) cP(vc) +c2

2H1L2 +O(3H1L2

).

The orbital stability of vc is then a onsequene of the onservation of E and P.

The argument extends to a well-prepared sum of solitons through a suient separation

of the ontributions of eah soliton.


The modulation parameters

In order to guarantee the two orthogonality onditions, we introdue modulation

parameters. Given > 0 and L > 0, we set

V(, L) ={

v NV(R), s.t. infak+1>ak+L

v Sc,a,sH1L2 < }

.

Lemma

There exist numbers 1

> 0, 1

> 0 and L1

> 0 suh that, given a solution

v C0([0,T ],H1(R) L2(R)) to (HLL), with v(, t) V(, L), for < 1

and L > L1

,

there exist two funtions a C1([0,T ],RN) and c C1([0,T ], (1, 1)N) suh that themap

(, t) = v(, t) Sc(t),a(t),s,

satises the orthogonality onditions

xvck (t),ak (t), (, t)

L2=

P

(vck (t),ak (t)), (, t)

L2= 0,

for any 1 k N and any t [0,T ]. Moreover, we have

(, t)H1L2 +N

k=1

|ck (t) c0

k | = O(),

N

k=1

(|ak (t) ck (t)|+ |c

k(t)|

)= O((, t)H1L2) +O(e

1

L), for t [0,T ].


Loalizing the ontributions of eah soliton

For small enough, we introdue the funtions 1

= 1, N+1 = 0, and

j (x) =1

2

(

1 + tanh(

(

x aj1(t) + aj (t)

2

)))

,

for 2 j N, and we set

F(v) = E(v)N

j=1

c0

j Pj (v), where Pj (v) =

R

(j j+1

)vw .

Lemma

There exist numbers > 0, 2

> 0, 2

> 0 and L2


v C0([0,T ],H1(R) L2(R)) to (HLL), with

v(, t) V(, L),

for < 2

and L > L2

, we have for t [0,T ]:

F(v) N

j=1

(E(vc0

j) c0j P(vc0

j))+

2

H1L2+O

( N

j=1

|cj (t) c0

j |2

)

+O(exp(

2

L)).

However, the funtional F(v) is no longer onserved along the ow.


The monotoniity formula

The onservation law for the momentum writes as

t(vw

)=

1

2

x

(

v2 + w2

(1 3v2

)+

3 v2

(1 v2)2(xv)

2

)

1

2

xxx ln(1 v2

).

Lemma

There exist numbers 3

> 0, 3

> 0 and L3


v C0([0,T ],H1(R) L2(R)) to (HLL), with

v(, t) V(, L),

for < 3

and L > L3

, we have

F (t) O(exp(

3

(L+ t)),

for any t [0,T ].

This allows us to obtain Theorem 2.


Perspetives

Asymptoti stability of the solitons (Yakine Bahri, PhD Thesis).

Constrution and orbital stability of solitons in higher dimension.

Asymptoti stability in higher dimension?


Thank you for your attention!


IntroductionThe LandauLifshitz equation

SolitonsThe anisotropic undamped Landau-Lifshitz equationConnection with the Schrdinger equation

Chain of solitons (D=1)Solitons and Multisolitons

The Cauchy problemOrbital stability of solitonsSketch of the proof of mains theoremsSketch of the proof of Theorem 1Sketch of the proof of Theorem 2

Perspectives and open problems

Stability in the energy space for chains of solitons of...

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