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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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