Geometric optics and boundary layers for Nonlinear Schrodinger Equations

Geometricopticsandboundarylayers
forNonlinear-Schro¨dingerEquations.
D.Chiron,F.Rousset
∗

Abstract
WejustifysupercriticalgeometricopticsinsmalltimeforthedefocusingsemiclassicalNon-
linearSchro¨dingerEquationforalargeclassofnon-necessarilyhomogeneousnonlinearities.The
caseofahalf-spacewithNeumannboundaryconditionisalsostudied.

1Introduction
WeconsiderthenonlinearSchro¨dingerequationinΩ
⊂
R
d
2εεΨ∂iε
+ΔΨ
ε
−
Ψ
ε
f
(
|
Ψ
ε
|
2
)=0
,
Ψ
ε
:
R
+
×
Ω
→
C
(1)
2t∂withanhighlyoscillatinginitialdatumundertheform
iεεεεΨ
|
t
=0
=Ψ
0
=
a
0
exp
ϕ
0
,
(2)
εwhere
ϕ
0
ε
isreal-valued.Weareinterestedinthesemiclassicallimit
ε
→
0.Thenonlinear
Schro¨dingerequation(1)appears,forinstance,inoptics,andalsoasamodelforBose-Einstein
condensates,with
f
(
ρ
)=
ρ
−
1,andtheequationistermedGross-Pitaevskiiequation,oralso
with
f
(
ρ
)=
ρ
2
(see[13]).Somemorecomplicatednonlinearitiesarealsousedespeciallyinlow
dimensions,see[12].
Atfirst,letusfocusonthecaseΩ=
R
d
.Toguesstheformallimit,when
ε
goestozero,itis
classicaltousethe
Madelungtransform
,i.etoseekforasolutionof(1)undertheform
i√Ψ
ε
=
ρ
ε
exp
ϕ
ε
.
εByseparatingrealandimaginarypartsanbyintroducing
u
ε
≡∇
ϕ
ε
,thisallowstorewrite(1)as
anhydrodynamicalsystem
∂
t
ρ
ε
+
∇
ρ
ε
u
ε
=0

ε
2

Δ
√
ρ
ε

(3)
ερ2

∂
t
u
ε
+
u
ε
∇
u
ε
+
∇
f
(
ρ
ε
)=
∇√
.
∗
LaboratoireJ.A.DIEUDONNE,Universite´deNice-SophiaAntipolis,ParcValrose,06108NiceCedex02,France,
chiron@unice.fr,frousset@unice.fr

1

Thesystem(3)isacompressibleEulerequationwithanadditionaltermintheright-handside
called
quantumpressure
.As
ε
tendsto0,thequantumpressureisformallynegligibleand(3)
reducestothe(compressible)Eulerequation

∂
t
ρ
+
∇
ρu
=0

(4)

∂
t
u
+
u
∇
u
+
∇
f
(
ρ
)=0
.
Thejustificationofthisformalcomputationhasreceivedmuchinterestrecently.Thecaseofanalytic
datawassolvedin[7].ThenfordatawithSobolevregularityandadefocusingnonlinearity,sothat
(4)ishyperbolic,itwasnoticedbyGrenier,[9],thatitismoreconvenienttousethetransformation
εϕΨ
ε
=
a
ε
exp
i
(5)
εandtoallowtheamplitude
a
ε
tobecomplex.Byusinganidentificationbetween
C
and
R
2
,this
allowstorewrite(1)as

ε
22
∂
t
a
ε
+
u
ε
∇
a
ε
+
a
∇
u
ε
=
εJ
Δ
a
ε
)6(
∂
t
u
ε
+

u
ε
∇

u
ε
+
∇

f
(
|
a
ε
|
2
)

=0
,
where
J
isthematrixofcomplexmultiplicationby
i
:
J
=0
−
1
.
01

When
ε
=0,wefindthesystem
a2
∂
t
a
+
u
∇
a
+
∇
u
=0
)7(∂
t
u
+
u
∇
u
+
∇
f
(
|
a
|
2
)=0
,

whichisanotherformof(4),sincethen(
ρ
≡|
a
|
2
,u
)solves(4).Therigorousconvergenceof(6)
towards(7)providedtheinitialconditionssuitablyconvergewasrigorouslyperformedbyGrenier
[9]inthecase
f
(
ρ
)=
ρ
(whichcorrespondstothecubicdefocusingNLS).Moreprecisely,itwas
provenin[9]thatthereexists
T>
0independentof
ε
suchthatthesolutionof(6)isuniformly
boundedin
H
s
on[0
,T
].IntermsoftheunknownΨ
ε
of(1),thisgivesthat
ϕε
∈
(0
,
1][0
,T
]
ε
H
supsup

Ψ
ε
exp
−
i

s
<
+
∞
forevery
s
where(
a,u
=
∇
ϕ
)isthesolutionof(7).Furthermore,thejustificationofWKB
expansionsundertheform
X
m

iϕiϕ
Ψ
ε
−
ε
k
a
k
e
ε
=
O
(
ε
m
)
e
ε
0=kforevery
m
wasperformedin[9].ThemainideaintheworkofGrenier[9]istousethesymmetrizer
11S
≡
diag1
,
1
,
4
f
′
(
|
a
|
2
)
,

,
4
f
′
(
|
a
|
2
)
2

ofthehyperbolicsystem(7)toget
H
s
energyestimateswhichareuniformin
ε
forthesingularly
perturbedsystem(6).Thecaseofnonlinearitiesforwhich
f
′
vanishesatzero(forinstancethe
case
f
(
ρ
)=
ρ
2
)wasleftopenedin[9].Theadditionaldifficultyisthatforsuchnonlinearities,the
system(7)isonlyweaklyhyperbolicat
a
=0andinparticularthesymmetrizer
S
becomessingular
at
a
=0.
Inmorerecentworks,see[19],[14],[1]itwasproventhatforeveryweaksolutionof(1)with
f
(
ρ
)=
ρ
−
1or
f
(
ρ
)=
ρ
,thelimitsas
ε
→
0
|
Ψ
ε
|
2
−
ρ
→
0in
L
∞
([0
,T
]
,L
2
)
ε
Im
Ψ¯
ε
∇
Ψ
ε
−
ρu
→
0in
L
∞
([0
,T
]
,L
l
1
oc
)(8)
holdundersomesuitableassumptionontheinitialdata.Theapproachusedinthesepapersis
completelydifferent,andreliesonthemodulatedenergymethodintroducedin[4].Theadvantage
ofthispowerfullapproachisthatitallowstodescribethelimitofweaksolutionsandtohandle
generalnonlinearitiesoncetheexistenceofaglobalweaksolutionintheenergyspacefor(1)
isknown.Nevertheless,itdoesnotgiveprecisequalitativeinformationonthesolutionof(1),
forexample,itdoesnotallowtoprovethatthesolutionremainssmoothonanintervaloftime
independentof
ε
iftheinitialdataaresmoothortojustifyWKBexpansionuptoarbitraryorders
insmoothnorms.
Inthework[2],thepossibilityofgettingthesameresultasin[9]forpurepowernonlinearities
f
(
ρ
)=
ρ
σ
inthecaseΩ=
R
d
wasstudied.Itwasfirstnoticedthat,thankstotheresultof[15],
athesystem

2
∂
t
a
+
∇
ϕ
∇
a
+Δ
ϕ
=0
)9(12
∂
t
ϕ
+
|∇
ϕ
|
2
+
f
(
|
a
|
2
)=0
,
withtheinitialcondition
a,ϕ
/t
=0
=
a
0
,ϕ
0
∈
H
∞
hasauniquesmoothmaximalsolution
(
a,ϕ
)
∈C
[0
,T
∗
[
,H
s
(
R
d
)
×
H
s
−
1
(
R
d
)forevery
s
.Itwasthenestablished:
Theorem1([2])
Let
d
≤
3
,
σ
∈
N
∗
andinitialdata
a
0
ε
,
ϕ
0
ε
≡
ϕ
0
in
H
∞
suchthat,forsome
functions
(
ϕ
0
,a
0
)
∈
H
∞
,
H
a
0
ε
−
a
0

s
=
O
(
ε
)
,
forevery
s
≥
0
.Then,thereexists
T
∗
>
0
suchthat
(9)
with
f
(
ρ
)=
ρ
σ
hasasmoothmaximal
solution
(
a,ϕ
)
∈C
([0
,T
∗
[
,H
∞
×
H
∞
)
.Moreover,thereexists
T
∈
(0
,T
∗
)
independentof
ε
,such
thatthesolutionof
(1)
,
(2)
remainssmoothon
[0
,T
]
andverifiestheestimate