The KdV KP I limit of the Nonlinear Schrodinger equation

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
+Δ
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
+Δ
⊥
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
=
cε
3
τ,X
1
=
x
=
ε
(
z
1
−
cτ
)
,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
+Δ
⊥
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
1ε
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
)