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The KdV KP I limit of the Nonlinear Schrodinger equation

De
34 pages
The KdV/KP-I limit of the Nonlinear Schrodinger equation D. Chiron? & F. Rousset.† Abstract We justify rigorously the convergence of the amplitude of solutions of Nonlinear-Schrodinger type Equations with non zero limit at infinity to an asymptotic regime governed by the Korteweg- de Vries equation in dimension 1 and the Kadomtsev-Petviashvili I equation in dimensions 2 and more. We get two types of results. In the one-dimensional case, we prove directly by energy bounds that there is no vortex formation for the global solution of the NLS equation in the energy space and deduce from this the convergence towards the unique solution in the energy space of the KdV equation. In arbitrary dimensions, we use an hydrodynamic reformulation of NLS and recast the problem as a singular limit for an hyperbolic system. We thus prove that smooth Hs solutions exist on a time interval independent of the small parameter. We then pass to the limit by a compactness argument and obtain the KdV/KP-I equation. 1 Introduction We consider the n-dimensional nonlinear Schrodinger equation i∂?∂? + 1 2∆z? = ?f(|?| 2) ? = ?(?, z) : R+ ? Rn ? C. (NLS) This equation is used as a model in nonlinear Optics (see for instance [19]) and in superfluidity and Bose-Einstein condensation (see, e.

  • assuming additional

  • ∂tv ?

  • energy space

  • kdv equation

  • time depen- dent

  • without assuming

  • kdv equation satisfying


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TheKdV/KP-IlimitoftheNonlinearSchro¨dingerequation

D.Chiron

&F.Rousset.

Abstract
WejustifyrigorouslytheconvergenceoftheamplitudeofsolutionsofNonlinear-Schro¨dinger
typeEquationswithnonzerolimitatinfinitytoanasymptoticregimegovernedbytheKorteweg-
deVriesequationindimension1andtheKadomtsev-PetviashviliIequationindimensions2
andmore.Wegettwotypesofresults.Intheone-dimensionalcase,weprovedirectlybyenergy
boundsthatthereisnovortexformationfortheglobalsolutionoftheNLSequationinthe
energyspaceanddeducefromthistheconvergencetowardstheuniquesolutionintheenergy
spaceoftheKdVequation.Inarbitrarydimensions,weuseanhydrodynamicreformulationof
NLSandrecasttheproblemasasingularlimitforanhyperbolicsystem.Wethusprovethat
smooth
H
s
solutionsexistonatimeintervalindependentofthesmallparameter.Wethenpass
tothelimitbyacompactnessargumentandobtaintheKdV/KP-Iequation.

1Introduction
Weconsiderthe
n
-dimensionalnonlinearSchro¨dingerequation
1Ψ∂i

z
Ψ=Ψ
f
(
|
Ψ
|
2
)Ψ=Ψ(
τ,z
):
R
+
×
R
n

C
.
(NLS)
2τ∂ThisequationisusedasamodelinnonlinearOptics(seeforinstance[19])andinsuperfluidityand
Bose-Einsteincondensation(see,
e.g.
[23],[10],[13]).
Weassumethat,forsome
ρ
0
>
0,
f
(
ρ
02
)=0,sothatΨ

ρ
0
isaparticularsolutionof(NLS).
WeareinterestedinsolutionsΨof(NLS)suchthat
|
Ψ
|≃
ρ
0
.Inthesequel,wetake
ρ
0
=1,the
1−generalcasefollowschangingΨforΨ˜

ρ
0
Ψand
f
for
f
˜(
R
)

f
(
ρ
02
R
).Then,fromnowon,we
considersmoothnonlinearities
f
∈C

(
R
,
R
)suchthat
f
(1)=0
,f

(1)
>
0(1)
andwillbeinterestedinsituationswhere
|
Ψ
|≃
1.Notethatthismeansthanksto(1)thatweshall
studytheequationinadefocusingregime.Atypicalexampleofnonlinearityissimply
f
(
R
)=
R

1
forwhich(NLS)istermedtheGross-Pitaevskiiequation.Equation(NLS)isanHamiltonianflow
associatedtotheGinzburg-Landautypeenergy(whenitmakessense)
1E
(Ψ)
≡|∇
z
Ψ
|
2
+
F
|
Ψ
|
2
dz,
Z2nRRwhere
F
(
R
)

2
f
(
r
)
dr
.
Z1∗
LaboratoireJ.A.DIEUDONNE,Universite´deNice-SophiaAntipolis,ParcValrose,06108NiceCedex02,France.
e-mail:
chiron@unice.fr

IRMAR,Universite´deRennes1,CampusdeBeaulieu,35042RennesCedex,France.
e-mail:
frederic.rousset@univ-rennes1.fr

1

1.1KdVandKP-IasymptoticregimesforNLS
Inasuitablescalingcorrespondingto
|
Ψ
|≃
1,thedynamicsfortheamplitudeofΨconverges,
indimension
n
=1,totheKorteweg-deVriesequation
12

t
v
+
kv∂
x
v

2

x
3
v
=0
,
(KdV)
c4andindimensions
n

2totheKadomtsev-Petviashvili-Iequation
13∂
x
2

t
v
+
kv∂
x
v

2

x
v


v
=0(KP-I)
c4where
v
=
v
(
t,X
)

R
,
X
=(
x,x

)

R
×
R
n

1
.Thecoefficients
c
and
k
arerelatedtothe
nonlinearity
f
by
c

f

(1)
>
0and
k

6+2
f
′′
(1)
.
(2)
2cpNotethattheKP-IequationreducestotheKdVequationif
v
doesnotdependon
x

.
Theformalderivationofthisregimeisasfollows.First,weconsiderasmallparameter
ε
,and
rescaletimeandspaceaccordingto
t
=

3
τ,X
1
=
x
=
ε
(
z
1


)
,X
j
=
ε
2
z
j
,j
∈{
2
,...,n
}
,
Ψ(
τ,z
)=
ψ
ε
(
t,X
)
.
(3)
Inthislongwaveasymptotics,thenonlinearSchro¨dingerequationfor
ψ
ε
readsnow
∂ψ
ε
ε
2
ε
4
icε
3

icε∂
x
ψ
ε
+

x
2
ψ
ε


ψ
ε
=
ψ
ε
f
(
|
ψ
ε
|
2
)
,X
=(
x,x

)

R
×
R
n

1
.
(4)
22t∂Weshallusethefollowingansatzfor
ψ
ε
ψ
ε
(
t,X
)=1+
ε
2
A
ε
(
t,X
)exp
iεϕ
ε
(
t,X
)(5)
wheretheamplitude
A
ε

R
isassumedtobeoforder1andtherealphase
ϕ
ε

R
isalsoassumed
tobeoforder1.Thisansatzisnaturalinthestabilityanalysisoftheparticularsolution
ψ
ε
=1
toslowlymodulatedperturbations(see[18],[19]).Wefocusonperturbationthattravelstothe
rightandareslowlymodulatedinthetransversedirectionthanksto(3).Importantsolutionsof
NLSthatariseinthisframeworkarethetravellingwaves.Theuseoftheansatz(5)tostudytheir
qualitativepropertiesisclassicalinthephysicslitterature.
Theansatz(3),(5)isadaptedsothatnonlinearanddispersiveeffectsarealloforderoneonthe
chosentimescale.NotethattheoccurenceoftheKdVorKP-Iequationasenveloppeequations
insuchregimesisexpected.Wereferforexampleto[2]andreferencesthereinforthederivation
oftheseequationsfromthewater-wavessystem.
Byplugging(5)in(4)andbyseparatingrealandimaginaryparts,wecanrewrite(4)asthe
system

2
c∂
t
A
ε

c∂
x
A
ε
+
ε
2

x
A
ε

x
ϕ
ε
+1+
ε
2
A
ε

x
2
ϕ
ε
+
ε
4


A
ε


ϕ
ε
22ε2
+1+
ε
2
A
ε
Δ

ϕ
ε
=0
)6(∂
2
A
ε
Δ
A
ε
ε
2

2
ε
4
21+
ε
2
A
ε
21+
ε
2
A
ε
22
ε
2
c∂
t
ϕ
ε

c∂
x
ϕ
ε

ε
2

x


ε
4



+

x
ϕ
ε
+
|∇

ϕ
ε
|
2
2ε
+1
f
(1+
ε
2
A
ε
)
2
=0
.
2

Now,assumingthat
A
ε

A
and
ϕ
ε

ϕ
as
ε

0,weformallyobtainfromthetwoequationsof
theabovesystemthat
1−
c∂
x
A
+

x
2
ϕ
=0
,

c∂
x
ϕ
+2
f

(1)
A
=0
.
(7)
2Notethatwehaveusedthat
f
(1)=0andthusthat
f
(1+
ε
2
A
ε
)
2

2
ε
2
f

(1)
A
atleadingorder.
1In(7)andfromthedefinition(2)of
c
,thefirstequationisjust

timesthederivativeofthe
c2secondequationwithrespectto
x
,hence,wehavefoundforthelimittheconstraint
2
cA
=

x
ϕ.
(8)
1Togetthelimitequationsatisfiedby
A
,wecanaddthefirstequationin(6)andtimesthe
c2derivativeofthesecondequationwithrespectto
x
inordertocancelthemostsingularterm.This
yieldstheequation
c∂
t
A
ε
+1

x
ϕ
ε

1

x

x
A
+11+
ε
2
A
ε
Δ

ϕ
ε
+
c∂
x
Q
(
ε
2
A
ε
)

2
ε

2
c
4
c
1+
ε
2
A
ε
2
ε
4
on2
x
4
c
2
c
+

x
A
ε

x
ϕ
ε
+1
A
ε

2
ϕ
ε
+1

x

(

x
ϕ
ε
)
2

+1

f

(1)+2
f
′′
(1)


x

(
A
ε
)
2

(9)
=4
c∂
x
1+
ε
2
A
ε

4
c∂
x
|∇

ϕ
|−
ε


A


ϕ,
ε
2

Δ

A
ε

ε
2

ε
2

2
εε
c
2
Q
(
r
)

f
(1+
r
)
2

2
f

(1)
r

f

(1)+2
f
′′
(1)
r
2
=
O
(
r
3
)
r

0
.
where

Stillonaformallevel,if
A
ε

A
and
ϕ
ε

ϕ
as
ε

0,thisyields
1122

t
A
+6+
2
f
′′
(1)
A∂
x
A

2

x
3
A


ϕ
=0
ihc2c4cbyusingtherelation(8).Consequently,wehaveobtainedthesytem


x
ϕ
=2
cA
hi
(10)
c
2
4
c
2
2
c

2

t
A
+6+2
f
′′
(1)
A∂
x
A

1

x
3
A
+1Δ

ϕ
=0
whichisareformulationoftheKP-Iequation.Notethatindimension1,
i.e.
when
n
=1,this
amountstoassumethatallthefunctionsinvolvedinthederivationdonotdependon
x

,thenthe
equationfor
A
in(10)justreducestotheKdVequationsinceΔ

ϕ
=0.
Finally,letusnoticethatbecauseofthescaling(3),forthesolutionΨoftheoriginal(NLS)
equationwithtime-scale1,theconvergencetoKdVorKP-Idynamicstakesplacefortimesoforder
3−.ε

Indimension
n
=1,theformalderivationoftheKdVequationfromthe(NLS)equationin
thisasymptoticregimeiswell-knowninthephysicsliterature(see,forexample,[18]),andisuseful
inthestabilityanalysisofdarksolitonsortravellingwavesofsmallenergy.Inthecaseofthe

3

Gross-Pitaevskiiequation,forinstance(thatisfor
f
(
R
)=
R

1),thetravellingwavesaresolutions
to(NLS)oftheformΨ(
τ,z
)=
U
(
z

στ
),sothat
U
solves
1−
iσ∂
z
U
+

zz
U
=
U
(
|
U
|
2

1)
,z

R
(11)
2withthecondition
|
U
|
(
z
)

1as
z
→±∞
.Forthisnonlinearity,explicitintegration(see,
e.g.
[26])
givesfor0
<σ<
1thenontrivialsolution
U
σ
(
z
)=
σ

i
1

σ
2
th
z
1

σ
2
.
ppInthisscaling,thespeedofsoundis1,hencethetravellingwavesaresubsonic.Inthetransonic
limit
σ

1,thusweset
σ
2
=1

ε
2
,
ε>
0small,andweobtain
2εsU
σ
(
z
)=


th(
εz
)+1

ε
2
=1

2
exp
iεϕ
ε
(
εz
)
,
pch(
εz
)
with
ϕ
ε
(
εz
)=

th(
εz
)+
O
(
ε
3
),andweseethatthiscorrespondstotheansatz(5)as
ε

0.
Furthermore,here,
A
ε
=

1
/
ch
2
doesnotdependon
ε
andisthesolitonoftheKdVequation
(
c
=1,
k
=6).Notethat(11)isalsooftenadimensionalizedintheform
2−
iσ∂
z
U
+

zz
U
=
U
(
|
U
|−
1)
.
√Inthiscase,thecriticalspeedonewhichisthesoundspeed,ischangedfor2.
Inhigherdimensions
n
=2,3,theconvergenceofthetravellingwavestotheGross-Pitaevskii
equation(
i.e.
(NLS)with
f
(
R
)=
R

1)withspeed

1toasolitonoftheKP-Iequationis
formallyderivedinthepaper[15],whilein[3],thisKP-Iasymptoticregimefor(NLS)indimen-
sion
n
=3isusedtoinvestigatethelinearinstabilityofthesolitarywavesofspeed

1.Onthe
mathematicallevel,indimension
n
=2,theconvergenceofthetravellingwavesofspeed

1for
theGross-PitaevskiiequationtoagroundstateoftheKP-Iequationisprovedin[5].

Here,weshallstudytherigorousderivationofKdV/KP-Ifrom(NLS)forarbitrarytimedepen-
dentsolutions.AllourresultsareinparticularvalidfortheGross-Pitaevskiiequation
f
(
R
)=
R

1.
When
n
=1,thereareglobalintimesolutionsof(NLS)intheenergyspaceandweshallprove
thatthesmallnessoftheenergypreventsΨfromhavingzeros.Thiswillallowustojustifythe
ansatz(5).Byusingtheconservationofenergyandmomentum,weshallgetdirectlythatifthe
initialdatumiswell-preparedinthesensethat
|

x
ϕ
0
ε

2
cA
0
ε
|
L
2
(
R
)

tendstozero(
cf.
(8)),thenwe
canpasstothelimitdirectlyonarbitrarilylargeintervaloftimefromtheglobalsolutionof(NLS)
intheenergyspacetowardsthesolutionofKdVintheenergyspacewithoutassumingadditional
regularityoftheinitialdata(Theorem3).Whenthedimensionislargerthanone,itdoesnotseem
possibletoprovethatifΨhasasmallenergythenΨdoesnotvanish.Asafirststep,todeal
withthehigherdimensionalcase,wethusneedtoprovedirectlythatthereindeedexistsasmooth
solutionof(4)undertheform(5)with(
A
ε

ε
)boundedin
H
s
for
s
sufficientlylargeonaninterval
oftimeindependentof
ε
.Thisexistenceresult(Theorem4)isestablishedbyrecasting(4)asan
hydrodynamicaltypesystemcloseto(6)butwithaparticularsingularPDElimitstructureasin
[20],[12],[24].Next,weshalljustifytheKdV/KP-Ilimitbyusingweakcompactnessarguments.
Forgeneralinitialdata(
i.e.
”ill-prepared”dataintheterminologyofsingularPDElimits),thatis

4

withoutassumingthat2
cA
ε


x
ϕ
ε
tendstozeroattheinitialtime(inordertobecompatiblewith
theconstraint(8)),weareabletopasstothelimit(Theorem5)inaweaksense:theamplitude
A
ε
convergestothesolutionoftheKdV/KP-Iequationweaklyin
L
2
([0
,T
]
×
R
n
).Ifthedataare
betterpreparedaccordingtotheconstraint(8),wecanjustifytheKdV/KP-Iasymptoticlimit
withstrongerconvergences,namelypointwiseintimeandglobalstronginspace(Theorem6).

1.2KdVasymptoticregimefor(NLS)intheenergyspace
Wefirstfocusonthedescriptionofourresultintheonedimensionalcase
n
=1,andworkonly
intheenergyspacefor(NLS)andthe
H
1
energyspaceforKdV.TheCauchyproblemfor(NLS)
isnotstandardbecauseoftheconditionatinfinity
|
Ψ
|→
1(see[9],[27],[8])whichisexpectedin
ordertogiveameaningtotheenergy
E
(Ψ).Wehavethefollowing:
Theorem1([27])
Thereexists
E
0
>
0
suchthatforevery
Ψ
0

H
l
1
oc
(
R
)
verifying
E

0
)
≤E
0
,
C
R
+
,H
1
(
R
)
.Moreover,
E
Ψ(
t
)=
E

0
)
for
t

0
.
a

nd
|
Ψ
0
|
(
z
)


1
as
|
z
|→

+

,

thereexistsauniquesolution
Ψ
to(NLS)suchthat
Ψ

Ψ
0

ThisTheoremisnotexactlyformulatedunderthisformin[27](TheoremIII.3.1).Nevertheless,
asweshallseeinLemma1,if
E
(Ψ)
≤E
0
issufficientlysmalland
|
Ψ
|→
1atinfinity,thenwecan
writeΨ=
ρe

with
|

x
ρ
|
L
2
(
R
)
+
|
ρ

1
|
L

(
R
)
+
|

x
φ
|
L
2
(
R
)
sufficientlysmallandhencewecanindeeduse[27](TheoremIII.3.1).
ItisalsoknownthattheCauchyproblemfortheKdVequation
1
[16]iswell-posedintheenergy
space:
Theorem2([16])
WeconsidertheCauchyproblemfortheKdVequation
132

t
v
+
kv∂
x
v

4
c
2

x
v
=0
,v
|
t
=0
=
v
0
.
If
v
0

H
1
(
R
)
,thenthereexistsauniquesolutionoftheKdVequationsatisfying
v
∈C
b
R
+
,H
1
(
R
)
and

x
v

L
l
4
oc
R
+
,L

(
R
)
.
Notethatitispossibletoprovethewell-posednessofKdVinspacesofmuchlowerregularitythan
H
1
(see[17]forexample)butweshallnotusetheseresultshere.
Ourfirstresultrelatesthesolutionof(NLS)obtainedinTheorem1inthescaling(3)andthe
solutionofKdVobtainedinTheorem2:
Theorem3(
n
=1)
Assumethat
(
A
0
ε
)
0
<ε<
1

H
1
and
(
ϕ
0
ε
)
0
<ε<
1

H
˙
1
enjoytheuniformesti-
1εεεmate
no
1<ε<0M

sup

A
0

H
1
(
R
)
+
ε


x
ϕ
0

2
cA
0

L
2
(
R
)
<
+

(12)
andthat
A
0
ε

A
0
in
L
2
(
R
)
as
ε

0
.
1
Here,itmighthappenthat
k
=0,inwhichcasetheKdVequationreducestotheso-called(linear)Airyequation
2

t
v

4
c
1
2

x
3
v
=0andtheCauchyproblemisthentrivialtosolve.

5

ψ
0
ε
=1+
ε
2
A
0
ε
exp
iεϕ
0
ε
(13)
Considertheinitialdatum

for
(4)
,andlet
ψ
ε

ψ
0
ε
+
C
R
+
,H
1
(
R
)
betheassociatedsolutionto
(4)
(givenbyTheorem1).
Then,thereexists
ε
0
>
0
,dependingonlyon
M
,suchthat,for
0


ε
0
,thereexisttwo
real-valuedfunctions
ϕ
ε
,
A
ε
∈C
(
R
+
×
R
,
R
)
suchthat
(
A
ε

ε
)
|
t
=0
=(
A
0
ε

0
ε
)
,and
ψ
ε
=1+
ε
2
A
ε
exp
iεϕ
ε
(14)
with
1+
ε
2
A
ε

21
.Furthermore,as
ε

0
,wehavetheconvergences
A
ε

A
in
C
[0
,T
]
,H
s
(
R
)
,∂
x
ϕ
ε

2
cA,
in
C
[0
,T
]
,L
2
(
R
)
forevery
s<
1
andevery
T>
0
,where
A
isthesolutionofKdVwithinitialvalue
A
0
.
Notethattheconvergenceholdsforarbitrarilylargeintervaloftimes[0
,T
].Moreover,letus
emphasizethattheinitialdataarewell-prepared(see(8))inthesensethat

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