(1)equation AndrédeLaire(UniversitéLille1) JointworkwithPhilippeGravejat(Éolepolytehnique) Shrödingerequationsandappliations CIRMJune18th2014 (2)(3)Thedynamisofmagnetizationinaferromagnetimaterialisgivenbythe LandauLifshitzequation

equation

AndrédeLaire(UniversitéLille1)

JointworkwithPhilippeGravejat(Éolepolytehnique)

Shrödingerequationsandappliations

CIRMJune18th2014

Thedynamisofmagnetizationinaferromagnetimaterialisgivenbythe

LandauLifshitzequation.

Themagnetizationisadiretioneld:

~

m

(x , t) : R ^D × R → S

⊂ R

,

~ = (m

, m

, m

)

|~

| = (m

+ m

+ m

)

^/

=

3Dferromagnetimaterials

2D:Thinmaterials.

1D:CesiumnikeltriuorideCsNiF3,theouplingbetweentheNiionsismuh

strongerthantheotherouplingsinthelattie.

∂ t

~ = −β ~

× ~

eff (

~ )

| {z }

precession

~

h

eff (

~ )

∂ t

~ = −β ~

× ~

eff (

~ )

| {z }

precession

−α~

× (

~ × ~

eff (~

))

| {z }

damping

~

h

eff (

~ )

α >

:

β >

α

+ β

=

~

eff (

~ ) = ∆

~ − λm

ˆ

∂ t

~ = −~

× (∆~

− λm

ˆ

),

~

m

(x , t) : R ^D × R → S

, ~

= (m

, m

, m

)

~

eff (

~ ) = ∆

~ − λm

ˆ

∂ t

~ = −~

× (∆~

− λm

ˆ

),

~

m

(x , t) : R ^D × R → S

, ~

= (m

, m

, m

)

∂ t m

= −m

(∆m

− λm

) + m

S∆m

∂ t m

= −m

∆m

− m

(∆m

+ λm

)

∂ t m

= −m

∆m

+ m

∆m

~

eff (

~ ) = ∆

~ − λm

ˆ

∂ t

~ = −~

× (∆~

− λm

ˆ

),

~

m

(x , t) : R ^D × R → S

, ~

= (m

, m

, m

)

Theequationishamiltonian. TheonservedHamiltonianistheLandau-Lifshitzenergy

E (~

(t)) :=

2

Z

R ^N

|∇

~ (x , t)|

dx + λ

2

Z

R ^N

m

(x , t)

dx = E (~

(

)), ∀t ∈ R .

Theanisotropyparameter

λ ≥

λ =

:

λ >

:

|x|→∞ m

(x, t) =

Inthesequel,weonsider

λ =

m

Whenthemap

m ˇ := m

+ im

doesnotvanish,itmaybewrittenas

ˇ m =

q

1

− m

3 exp

iϕ.

Thefuntions

v = m

and

w = ∇ϕ



 

 

∂ t v = −

(

− v

)w ,

∂ t w = ∇

v (|w |

−

) + ∆v

1

− v

+ v |∇v |

(

− v

)

.

(HLL)

Thelinearizedequationaroundthezerosolutionisdispersivewithdispersionrelation

ω

= |k|

+ |k|

.

Dispersionveloityisatleastequaltothesoundspeed

c s =

Welookfortravelingwavessolutionspropagatingalongthe

x

-axiswithspeed

c

~

m

(x ) = ~

(x

− ct , x

, . . . , x D ).

Thentheprole

~

−c ∂

~

+ ~

× (∆~

− u

ˆ

) =

.

Taking

~

×

~

u

× (~

× ∆~

) = ~

(~

· ∆~

) − ∆~

(~

· ~

) = −~

|∇~

|

− ∆~

,

weobtain

− ∆ ~

= |∇~

|

~

+ u

~

− u

ˆ

+ c~

× ∂

~

.

c

Welookfortravelingwavessolutionspropagatingalongthe

x

-axiswithspeed

c

~

m

(x ) = ~

(x

− ct , x

, . . . , x D ).

Thentheprole

~

−c ∂

~

+ ~

× (∆~

− u

ˆ

) =

.

Taking

~

×

~

u

× (~

× ∆~

) = ~

(~

· ∆~

) − ∆~

(~

· ~

) = −~

|∇~

|

− ∆~

,

weobtain

− ∆ ~

= |∇~

|

~

+ u

~

− u

ˆ

+ c~

× ∂

~

.

c

Trivialsolutions: onstantsin

S

× {

}

Itisenoughtoonsider

c ≥

propertiesfor

D =

,

subsonispeeds

|c | <

LinandWei(2010)provedtheexisteneofsmallspeedtravellingwaveswitha

vortex-antivortexpairwhen

D =

d.L.(2014): thenon-existeneofnon-onstantsubsonismallenergytravelling

waves for

D =

,

,

d.L.(2014): alsosmoothnessandtheiralgebraideayatinnityfor

D ≥

Somenumerial resultsfor

D =

0 15 30 45 60

15 30

p E

c → 0 c → 1

E = p

c≈ 0 . 78

Somenumerial resultsfor

D =

u

for

c =

.

Somenumerial resultsfor

D =

c =

.

c =

.

Let

~

c

ψ = u

+ iu

1

+ u

,

wehavethat

ψ

∆ψ +

− |ψ|

1

+ |ψ|

ψ + ic∂

ψ =

ψ ¯

1

+ |ψ|

(∇ψ · ∇ψ),

whihseemslikeaperturbedequationforthetravelingwavesforaNonlinearShrödinger

equation(GrossPitaevskiiequation).

Let

~

c

ψ = u

+ iu

1

+ u

,

wehavethat

ψ

∆ψ +

− |ψ|

1

+ |ψ|

ψ + ic∂

ψ =

ψ ¯

1

+ |ψ|

(∇ψ · ∇ψ)

thatseemslikeaperturbedequationforthetravelingwavesforanonlinearShrödinger

equation(GrossPitaevskiiequation):

∆Ψ + (

− |Ψ|

)Ψ − ic ∂

Ψ =

.

Solitonsfor

D =

Indimensionone,thetravelingwavesolutionsarethefollowingexpliitdarksolitons.

Lemma(The one-dimensionalase)

Let

D =

c ≥

~

c

Then0

≤ c <

u

= c

( p

1

− c

x ), u

=

( p

1

− c

x), u

= p

1

− c

( p

1

− c

x).

Multisolitonsfor

D =

TheLandau-Lifshitzequationisintegrablebymeansoftheinversesatteringmethod. In

partiular,thereareexpliitformulaeformulti-solitons. Theybehavelikeasumof

orderedsolitonsas

t → ∞

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Multisolitonsfor

D =

TheLandau-Lifshitzequationisintegrablebymeansoftheinversesatteringmethod. In

partiular,thereareexpliitformulaeformulti-solitons. Theybehavelikeasumof

orderedsolitonsas

t → ∞

(Loading...)

For

c 6=

u ˇ =

− u

2

exp

i ϕ

v c = v c , w c

= u

, ∂ x ϕ ,

where

v c (x) = (

− c

)

osh

((

− c

)

x ,

and

w c (x) = c v c (x )

1

− v c (x )

.

For

c 6=

u ˇ =

− u

2

exp

i ϕ

v c = v c , w c

= u

, ∂ x ϕ ,

where

v c (x) = (

− c

)

osh

((

− c

)

x ,

and

w c (x) = c v c (x )

1

− v c (x )

.

Ahainofsolitonsisdenedasaperturbationofasumofsolitons

S c , a , s (x ) = X N

j=

s j v c _j (x − a j ) = X N

j=

s j v c _j (x − a j ), s j w c _j (x − a j ) ,

with

a = (a

, . . . , a N ) ∈ R ^N

c = (c

, . . . , c N ) ∈ (−

,

) ^N

s = (s

, . . . , s N ) ∈ {±

} ^N

Theenergyspaeisdenedas

E ( R ) =

m : R → S

,

.

. m ^′ ∈ L

( R )

m

∈ L

( R ) .

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

Let

m

∈ E ( R )

m ∈ L ^∞ ( R , E( R ))

∂ t m + m × (∂ xx m − m

e

) =

,

withinitialdatum

m

Remark: nouniqueness

Theorem(Existeneanduniquenessofsmoothsolutions)

Let

k ≥

m

∈ E ( R )

[m

] ^′ ∈ H ^k ( R )

m : R × R → S

m

Inthehydrodynamiframework, theLandau-Lifshitzenergyisequalto

E (v , w ) =

2

Z

R

(∂ x v )

1

− v

+ (

− v

)w

+ v

.

Thenon-vanishingspaeisdenedas

N V( R ) =

v = (v , w ) ∈ H

( R ) × L

( R ),

.

.

x∈R |v(x)| <

.

Anotheronservedquantityisthemomentum:

P(v , w ) = Z

R

vw.

Theorem1(d.L-Gravejat, 2014)

Let

v

= (v

, w

) ∈ N V( R )

(i )

T

>

v ∈ C

([

, T

), N V( R ))

(HLL)withinitialdatum

v

v n ∈ C ^∞ ( R × [

, T ])

to(HLL)satisfying

v n → v

C

([

, T ], N V( R )),

n → ∞,

T < T

.

For

c 6=

v c

E

(p c ) =

E(v ), v : R → C

.

. P(v ) = p c .

Thespeed

c

p c

p c =

(

− c)

c

.

TheEuler-Lagrangeequationisy

E ^′ (v c ) = cP ^′ (v c ).

E

E = p

2

Theorem2(d.L.-Gravejat 2014)

Let

c

∈ (−

,

) ^N

s

∈ {±

} ^N

v

= (v

, w

) ∈ N V( R )

c

< · · · <

< · · · < c _N

.

Thereexistfournumbers

α ^∗ >

ν ^∗ >

A ^∗ >

L ^∗ >

v

− S c

,a

,s

H

×L

= α

< α ^∗ ,

forpositions

a

∈ R ^N

a _k+

− a

_k ,

≤ k ≤ N −

= L

> L ^∗ ,

thenthereexistauniquesolution

v ∈ C

( R ₊ , N V( R ))

v

wellas

N

a k ∈ C

( R ₊ , R )

a k (

) = a

_k

a ^′ _k (t) − c _k

≤ A ^∗ α

+ e ^{−ν ∗ L}

,

Corollary3(d.L-Gravejat2014)

Let

c

∈ (−

,

) ∪ (

,

)

a

∈ R

s

∈ {±

}

v

∈ N V( R )

α ^∗ >

A ^∗ >

v

− s

v c

(· − a

)

H

×L

< α ^∗ ,

thenthereexistauniquesolution

v ∈ C

( R ₊ , N V( R ))

v

wellasafuntion

a ∈ C

( R ₊ , R )

a(

) = a

a ^′ (t) − c

≤ A ^∗ α ^∗ ,

and

v(·, t) − s

v c

(· − a(t))

H

×L

≤ A ^∗ α ^∗ ,

forany

t ∈ R ₊

Itfollowsfromtheexisteneofsmoothsolutionsto(LL).

Lemma

Let

k ≥

N V ^k ( R ) =

v = (v , w ) ∈ H ^k+

( R ) × H ^k ( R ),

.

.

x∈R |v(x)| <

.

Givenapair

v

∈ N V ^k ( R )

T

>

v ∈ L ^∞ ([

, T

), N V ^k ( R ))

v

T

is

haraterizedbytheondition

lim

t→T

x∈R |v (x , t)| =

,

T

< +∞.

Givenasmoothsolution

v = (v , w )

Ψ =

2

∂ x v

(

− v

)

+ i (

− v

)

w

exp

i θ,

wherethe phase

θ

θ(x , t) = −

Z x

−∞

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

Wealsoset

F (v , Ψ)(x ,t) = Z x

−∞

v(y, t) Ψ(y, t) dy.

Themap

Ψ

i ∂ t Ψ + ∂ xx Ψ+

|Ψ|

Ψ +

2

v

Ψ

−

Ψ(

−

F (v , Ψ))

(

−

F (v , Ψ)) =

,

whilethefuntion

v

owmaporrespondingtothissystem.

Lemma

Let

T ^∗ >

(v

, Ψ

)

(v

, Ψ

)

withinitialdatum

(v j

, Ψ

j )

τ >

k v _j

k L

kΨ

j k L

K

v

− v

C

([

,T],L

) +

Ψ

− Ψ

C

([

,T],L

)

≤ K v

− v

L

+ Ψ

− Ψ

L

,

forany

T ∈ [

,

{τ, T _∗ }]

Theorem1derivesfromexpressingthisontinuitypropertyintermsofthepair

v = (v , w )

ThestrategyissimilartotheonedevelopedbyMartel-Merle-Tsai'02,'05,forthe

Korteweg-deVriesandthenonlinearShrödingerequations(seealso

Béthuel-Gravejat-Smets'12).

Itresultsfromthefollowingquantiationoftheminimizingnatureofasoliton.

Lemma

Let

c ∈ (−

,

) ∪ (

,

)

H c = E ^′′ (v c ) − cP ^′′ (v c )

Λ c >

suhthat

h H c (ε), εi L

≥ Λ c kεk

H

×L

,

foranypair

ε ∈ H

( R ) × L

( R )

h∂ x v c , εi L

= hP ^′ (v c ), εi L

=

.

When

ε = v − v c

E (v) − cP(v)= E (v c ) − cP(v c ) +

2

hH c (ε), εi L

+ O kεk

H

×L

Λ

Inordertoguaranteethetwoorthogonalityonditions,weintroduemodulation

parameters. Given

α >

L >

V(α, L) = n

v ∈ N V( R ),

.

.

a _k+

1

>a _k +L kv − S c ∗ , a , s ∗ k H

×L

< α o .

Lemma

Thereexistnumbers

α

>

ν

>

L

>

v ∈ C

([

, T ], H

( R ) × L

( R ))

v(·, t) ∈ V(α, L),

α < α

L > L

thereexisttwofuntions

a ∈ C

([

, T ], R ^N )

c ∈ C

([

, T ], (−

,

) ^N )

map

ε(·, t) = v(·, t) − S c(t),a(t),s ^∗ ,

satisestheorthogonalityonditions

∂ x v c _k (t),a k (t) , ε(·, t)

L

=

P ^′ (v c _k (t),a k (t) ), ε(·, t)

L

=

,

forany1

≤ k ≤ N

t ∈ [

, T ]

kε(·, t)k H

×L

+

X N

k=

|c k (t) − c k

| = O(α),

For

ν

φ

=

φ N+

=

φ j (x) =

2

1

+

ν

x − a _j−

(t) + a j (t)

2

,

for2

≤ j ≤ N

F(v) = E (v) −

X N

j=

c j

P j (v),

P j (v) = Z

R

φ j − φ j+

vw.

Lemma

Thereexistnumbers

Λ >

α

>

ν

>

L

>

v ∈ C

([

, T ], H

( R ) × L

( R ))

v(·, t) ∈ V(α, L),

for

α < α

L > L

t ∈ [

, T ]

Theonservationlawforthemomentumwritesas

∂ t vw

=−

2

∂ x

v

+ w

−

v

+

− v

(

− v

)

(∂ x v )

−

2

∂ xxx

− v

.

Lemma

Thereexistnumbers

α

>

ν

>

L

>

v ∈ C

([

, T ], H

( R ) × L

( R ))

v(·, t) ∈ V(α, L),

for

α < α

L > L

,wehave

F ^′ (t) ≤ O

(−ν

(L + t) ,

forany

t ∈ [

, T ]

Asymptotistabilityofthesolitons(YakineBahri,PhDThesis).

Construtionandorbitalstabilityofsolitonsinhigherdimension.

Asymptotistabilityinhigherdimension?