WKB Analysis for the Gross-Pitaevskii Equation With Non-Trivial Boundary Conditions at Infinity

www.elsevier.com/locate/anihpc

WKB analysis for the Gross–Pitaevskii equation with non-trivial boundary conditions at infinity

Thomas Alazard

, Rémi Carles

aCNRS & Université Paris-Sud, Mathématiques UMR CNRS 8628, Bât. 425, 91405 Orsay cedex, France

bCNRS & Université Montpellier 2, Mathématiques UMR CNRS 5149, CC 051, Place Eugène Bataillon, 34095 Montpellier cedex 5, France Received 4 October 2007; accepted 8 February 2008

Available online 7 May 2008

Abstract

We consider the semi-classical limit for the Gross–Pitaevskii equation. In order to consider non-trivial boundary conditions at infinity, we work in Zhidkov spaces rather than in Sobolev spaces. For the usual cubic nonlinearity, we obtain a point-wise description of the wave function as the Planck constant goes to zero, so long as no singularity appears in the limit system. For a cubic-quintic nonlinearity, we show that working with analytic data may be necessary and sufficient to obtain a similar result.

©2008 Elsevier Masson SAS. All rights reserved.

MSC:35B40; 35C20; 35Q55; 37K05; 37L50; 81Q20; 82D50

Keywords:Semi-classical analysis; Gross–Pitaevskii equation; Zhidkov spaces

1. Introduction

We study the semi-classical limith¯→0 for the Gross–Pitaevskii equation ih∂¯ tu+ ¯h²

2mu=V u+f

|u|² u,

wherex∈Rⁿ. In the case of Bose–Einstein condensation (BEC), the external potentialV =V (t, x)models an external trap, and the nonlinearityf describes the non-linear interactions of the particles (see e.g. [11,25,19]). We consider two types of nonlinearityf (after renormalization):

• Cubic nonlinearity:f (|u|²)u=(|u|²−1)u.

• Cubic-quintic nonlinearity:f (|u|²)u=(|u|⁴+λ|u|²)u,λ∈R.

The cubic nonlinearity is certainly the most commonly used model in BEC. The defocusing nonlinearity corresponds to a positive scattering length, as in the case of ⁸⁷Rb, ²³Na and ¹H. Note that this model is also used in superfluid

* Corresponding author.

E-mail addresses:Thomas.Alazard@math.cnrs.fr (T. Alazard), Remi.Carles@math.cnrs.fr (R. Carles).

0294-1449/$ – see front matter ©2008 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2008.02.006

theory. See e.g. [11,25,19] and references therein. The cubic-quintic nonlinearity, which is mostly used as an envelope equation in optics, is also considered in BEC for alkalimetal gases (see e.g. [14,1,24]), in which caseλ <0. The cubic term corresponds to a negative scattering length, and the quintic term to a repulsive three-body elastic interaction. We also consider the caseλ >0 (positive scattering length).

1.1. Cubic nonlinearity

Up to rescaling the Planck constant, we consider the limitε→0 for:

iε∂_tu^ε+ε²

2u^ε=V u^ε+

|u^ε|²−1

u^ε, x∈Rⁿ, n1, (1.1)

u^ε(0, x)=a₀^ε(x)e^iφ0(x)/ε. (1.2)

Our initial data do not necessarily decay to zero at infinity. Typically, we do not assumea₀^ε∈L²(Rⁿ)(see Theorem 1.3 below). Recently, the Cauchy problem [12,17] and the semi-classical limit [21] for (1.1) withV ≡0 have been studied more systematically. When the external potentialV is zero,V ≡0, the Hamiltonian structure yields, at least formally:

d

dtε∇u^ε(t )²

L²+u^ε(t )²−1²

L²

=0.

In this case, a natural space to study the Cauchy problem associated to (1.1) is the energy space (see e.g. [6,17] and references therein)

E=

u∈H_loc¹ Rⁿ

; ∇u∈L² Rⁿ

, |u|²−1∈L² Rⁿ

.

For this quantity to be well defined, one cannot assume thatu^εis inL²(Rⁿ); morally, the modulus ofu^εgoes to one at infinity. To study solutions which are bounded, but not inL²(Rⁿ), P.E. Zhidkov introduced in the one-dimensional case in [29] (see also [30]):

X^s Rⁿ

=

u∈L^∞ Rⁿ

; ∇u∈H^s−1 Rⁿ

, s > n/2. (1.3)

We also denote X^∞:=

s>n/2X^s. The study of these spaces was generalized in the multidimensional case by C. Gallo [12]. They make it possible to consider solutions to (1.1) whose modulus has a non-zero limit as|x| → ∞, but not necessarily satisfying|u^ε(t,·)|²−1∈L²(Rⁿ). We shall also use these spaces.

Recently, P. Gérard [17] has solved the Cauchy problem for the Gross–Pitaevskii equation in the more natural spaceE, in space dimensions two and three. The main novelty consists in working with distances instead of norms, in order to apply a fixed point argument inE. In particular, the constraint|u^ε(t,·)|²−1∈L²(Rⁿ)is satisfied.

To our knowledge, if the initial data do not vanish at infinity, the introduction of an (unbounded) external potential in Gross–Pitaevskii equation has no physical motivation. Note also that ifV is an harmonic potential, then the formal Hamiltonian corresponding to (1.1) is necessarily infinite (see Section 2.3). On the other hand, introducing a quadratic external potential or considering a quadratic initial phaseφ₀makes no difference in our analysis. The model (1.1)–(1.2) withV ≡0 andφ₀quadratic is certainly more physically relevant, and does not seem to enter into the framework of the previous mathematical studies. Another motivation to introduce this external potential stems from the study of the semi-classical limit of the Schrödinger–Poisson system, where|u^ε|²−1 is replaced withV_p^ε given byV_p^ε= q(|u^ε|²−c). This models appears in the semi-conductor theory where the real numberqmodels a charge, which we may take equal to one here, and the functionc=c(x)models a doping profile, which we may take to bec≡1. As in [2], we will prove that ifV grows quadratically in space, then if|u^ε(t=0,·)|²−1∈L²(Rⁿ), one must not expect

|u^ε(t,·)|²−1∈L²(Rⁿ)fort >0.

Assumptions.We assume that the potential and the initial phase are of the form:

• V ∈C^∞(Rt×Rⁿx), andV =Vquad+Vlin, whereVquad(t, x)=^txM(t )xis a quadratic form, withM(t )∈Sn(R) a symmetricn×nmatrix, depending smoothly ont, and∇Vlin∈C^∞(Rt;X^s)for alls > n/2.

• φ0∈C^∞(Rⁿ), andφ0=φquad+φlin, whereφquad(x)=^txQ0x is a quadratic form, withQ0a symmetric matrix inMn×n(R), and∇φlin∈X^∞.

Note that our assumptions include the case whereV_lin andφ_lin are linear in x. In general, these functions are sub-linear inx, since their gradient is bounded.

Lemma 1.1.There existT >0and a unique solutionφ_eik∈C^∞([0, T] ×Rⁿ)to:

∂_tφ_eik+1

2|∇xφ_eik|²+V_quad=0; φ_eik|t=0=φ_quad. (1.4)

Moreover,φeikis a quadratic form inx:

φ_eik(t, x)=^txQ(t )x, (1.5)

whereQ(t )∈Sn(R)is a smooth function oft.

Proof. Existence and uniqueness follow from [9, Lemma 1]. To prove thatφ_eik is quadratic inx, seekφ_eik of the form (1.5). Then (1.4) is equivalent to the system of ordinary differential equations

Q(t )˙ +2Q(t )²+M(t )=0; Q(0)=Q₀.

The lemma then follows from Cauchy–Lipschitz Theorem. 2

Remark 1.2.As in [2], we shall use the following geometrical interpretation of the above lemma. The timeT is such that fort∈ [0, T], the map given by

∂_tx(t, y)= ∇xφ_eik

t, x(t, y)

=Q(t )x(t, y); x(0, y)=y,

defines a global diffeomorphism onRⁿ. Therefore, the characteristics associated to the operator∂t+ ∇φeik· ∇do not meet fort∈ [0, T], and this operator is a smooth transport operator:

(∂_tf + ∇φeik· ∇f )

t, x(t, y)

=∂_t f

t, x(t, y) . Note that ifQ(t )and its anti-derivative commute, then we have

x(t, y)=exp t 0

Q(τ ) dτ

y.

Theorem 1.3.Suppose that there exista₀, a₁∈X^∞such that:

a₀^ε−a₀−εa₁X^s =o(ε), ∀s > n/2. (1.6)

There existT_∗∈ ]0, T]independent ofε∈ ]0,1], and a unique solutionu^ε∈C^∞∩L^∞([0, T_∗] ×Rⁿ)to(1.1)–(1.2).

Moreover, there exista, φ∈C^∞([0, T_∗] ×Rⁿ)witha,∇φ∈C([0, T_∗];X^s)for alls > n/2, such that:

lim sup

ε→0

u^ε(t,·)−a(t,·)e^{i(φ (t,·)+φeik(t,·))/ε}

L∞(Rⁿ)=O(t ) ast→0. (1.7)

The functionsa and φdepend non-linearly onφ₀and a₀ (see(3.1)below). There existsφ⁽¹⁾∈L^∞([0, T_∗] ×Rⁿ), real-valued, with∇φ⁽¹⁾∈C([0, T_∗];X^s)for alls > n/2, such that:

lim sup

ε→0

u^ε−ae^iφ(1)e^{i(φ+φeik)/ε}

L^∞([0,T_∗]×Rⁿ)=0. (1.8)

The modulationφ⁽¹⁾is a non-linear function ofφ₀,a₀anda₁(see(3.2)below).

Remark 1.4.Several applications of this general results are given, in Sections 3, 4 and 5.

Remark 1.5.If we assume moreover

a₀^ε−a0−εa1X^s =O(ε²), ∀s > n/2, then the above error estimate can be improved:

u^ε−ae^iφ(1)e^{i(φ+φeik)/ε}

L^∞([0,T_∗]×Rⁿ)=O(ε).

Remark 1.6. The above result and system (3.2) below show that in general, it is necessary to know the initial am- plitudea^ε₀ up to the order o(ε)to describe the leading order behavior of thewave function u^ε. It is not necessary to knowa^ε₀ with such precision to study the convergence of quadratic observables. See Section 6. In particular, in Theorem 6.1, we extend the result of [21] to the three-dimensional case (on a bounded domain, or outside a bounded domain).

Remark 1.7. Most of the results that we present here remain valid in a space-periodic setting, that is if we as- sume x ∈ Tⁿ. In that case, compactness arguments show that the proof of Theorem 1.3 remains valid when V ∈C^∞(Rt ×Tⁿx)andφ0∈C^∞(Tⁿ). On the other hand, the discussions in Sections 2.3 and 5 become irrelevant on the torus. Finally, note that it is equivalent to work in Sobolev spaces, sinceX^s(Tⁿ)=H^s(Tⁿ)fors > n/2.

The analysis detailed in Sections 2 and 3 shows that the formal part of [8] can be justified in the present framework.

We shall only state a typical consequence of this approach:

Corollary 1.8 (Instability). Let n 1, a0, a1∈ C^∞ ∩X^∞(Rⁿ), with Re(a¯0a1)≡0, and φ0 ∈C^∞(Rⁿ), with

∇φ0∈X^∞. Letu^εandv^εsolve the initial value problems:

iε∂tu^ε+ε²

2u^ε=

|u^ε|²−1

u^ε; u^ε|t=0=a0e^iφ0/ε, iε∂_tv^ε+ε²

2 v^ε=

|v^ε|²−1

v^ε; v^ε|t=0=(a₀+δ^εa₁)e^iφ0/ε,

where δ^ε →0. Assume that there exists N ∈N such that δ^ε/ε^1−N1 → +∞. Then we can findt^ε→0 such that lim infε→0u^ε(t^ε)−v^ε(t^ε)L^∞>0. In particular,

lim inf

ε→0

u^ε−v^εL^∞([0,t^ε]×Rⁿ)

u^ε_|t=0−v_|^ε_t=0L^∞(Rⁿ) = +∞. Remark 1.9.Note that ifφ₀≡0, then we also have:

lim inf

ε→0

u^ε−v^εL^∞([0,t^ε]×Rⁿ)

u^ε_|t=0−v^ε_|t=0X^s = +∞, ∀s > n/2.

This shows that the instability mechanism is not due to regularity issues. It is due to the fact that (1.1) is super-critical as far as WKB analysis is concerned: the small initial perturbation (of orderδ^ε) yields a high-frequency perturbation of the evolution (a multiplicative factor of the forme^{−2it δεRe(a¯0a1)/ε}).

1.2. Cubic-quintic nonlinearity

Denotef_λ(y)=y²+λy. We now consider

⎧⎨

⎩

iε∂tu^ε+ε²

2u^ε=fλ

|u^ε|²

u^ε, x∈Rⁿ, n1, u^ε(0, x)=a₀^ε(x)e^iφ0(x)/ε.

(1.9) Note that in (1.9), we assume that there is no external potential,V =0. We also assume that there is no initial quadratic oscillation:φ0∈C^∞(Rⁿ;R), with∇φ0∈X^∞. The caseλ >0,V =0, witha₀^ε∈H^∞, is contained in [9]. We assume Vquad=0 here in order to consider non-zero boundary conditions at infinity. We also assumeVlin=0 for simplicity only.

Plugging an approximate solution of the formu^ε≈ae^iφ/ε, withaandφindependent ofε, and passing to the limit ε→0 as in [15,21], we find formally that(ρ, v):=(|a|²,∇φ)solves:

∂_tρ+div(ρv)=0,

∂tv+v· ∇v+ ∇ fλ(ρ)

=0. (1.10)

Ifλ >0, then the problem is hyperbolic. Essentially, the result of Theorem 1.3 remains valid. Whenλ <0, the above problem is hyperbolic forρ >|λ|/2 and elliptic forρ <|λ|/2. This feature is reminiscent of Euler equations of gas dynamics in Lagrangian coordinates:

∂_tu+∂_xv=0,

∂tv+∂x

p(u)

=0. (1.11)

As recalled in [23], a typical mathematical example for van der Waals state laws is given byp(u)=(u²−1)u. The problem is hyperbolic ifu >1/√

3, and elliptic ifu <1/√

3. Hadamard’s argument implies that the only reasonable framework to study (1.10) or (1.11) is that of analytic functions (see [23]). In this case, we refer to the approach of [16,28]. More details are given in Section 7. When the elliptic region for (1.10) is avoided, then essentially, Theo- rem 1.3 remains valid:

Theorem 1.10.Suppose that there exista₀, a₁∈X^∞such that:

a₀^ε−a₀−εa₁

X^s =o(ε), ∀s > n/2.

Assume moreover thatφ₀∈C^∞(Rⁿ;R)with∇φ₀∈X^∞, and:

• Eitherλ >0,

• Orλ <0and there existsδ >0such that|a₀(x)|²δ+^|λ₂^|,∀x∈Rⁿ.

Then there existε_∗, T_∗>0, and a unique solutionu^ε∈C^∞∩L^∞([0, T_∗] ×Rⁿ)to(1.9)for allε∈ ]0, ε_∗]. Moreover, there exista, φ∈C^∞([0, T_∗] ×Rⁿ)witha,∇φ∈C([0, T_∗];X^s)for alls > n/2, such that:

lim sup

ε→0

u^ε(t,·)−a(t,·)e^{iφ (t,·)/ε}

L^∞(Rⁿ)=O(t ) ast→0.

There existsφ⁽¹⁾∈L^∞([0, T_∗] ×Rⁿ), real-valued, with∇φ⁽¹⁾∈C([0, T_∗];X^s)for alls > n/2, such that:

lim sup

ε→0

u^ε−ae^iφ(1)e^iφ/ε

L^∞([0,T_∗]×Rⁿ)=0.

1.3. Structure of the paper

In Section 2, we construct the solutionu^ε asu^ε=a^εe^iΦε/ε, wherea^ε is complex-valued andΦ^ε is real-valued.

This yields the existence part of Theorems 1.3 and 1.10. The proof of these theorems is completed in Section 3, where the limit of(a^ε, Φ^ε)asεgoes to zero is studied. We give three examples of applications of Theorem 1.3 in Section 4, in the caseφeik=0. In Section 5, we study the time evolution of a non-trivial boundary condition at infinity when φ_eik=0. In Section 6, we investigate the limit of the position and current densities. Finally, we explain why working in an analytic setting is often necessary (and always sufficient) in the case of the cubic-quintic nonlinearity.

2. Construction of the solution

2.1. Phase-amplitude representation: the caseφeik=V =0

When V and φ₀ are identically zero, the existence and uniqueness part of Theorem 1.3 was established by C. Gallo [12]. Note however that with our scaling, the fact that T_∗ is independent of ε∈ ]0,1] does not follow from [12]. Since the approach in Zhidkov spaces is rather similar to the one in Sobolev spaces, we shall essentially explain the new aspects of the proof. To treat both cubic and cubic-quintic nonlinearities, consider the general equation

⎧⎨

⎩

iε∂_tu^ε+ε²

2u^ε=f

|u^ε|²

u^ε, x∈Rⁿ, n1, u^ε(0, x)=a^ε₀(x)e^iφ0(x)/ε,

(2.1)

where f ∈C^∞(R₊;R). We keep the hierarchy introduced by E. Grenier [18]: seek u^ε=a^εe^iΦε/ε, where a^ε is complex-valued, andΦ^εis real-valued. We impose

⎧⎪

⎨

⎪⎩

∂_tΦ^ε+1

2|∇Φ^ε|²+f

|a^ε|²

=0; Φ^ε|t=0=φ₀,

∂_ta^ε+ ∇Φ^ε· ∇a^ε+1

2a^εΦ^ε=iε

2a^ε; a^ε|t=0=a₀^ε.

(2.2)

As an intermediary unknown function, introduce the “velocity”v^ε= ∇Φ^ε. Separate real and imaginary parts ofa^ε, a^ε=a₁^ε+ia^ε₂, and introduce:

u^ε=

⎛

⎜⎜

⎜⎜

⎜⎝ a₁^ε a₂^ε v₁^ε ... v_n^ε

⎞

⎟⎟

⎟⎟

⎟⎠

, u^ε₀=

⎛

⎜⎜

⎜⎜

⎜⎝ Re(a₀^ε) Im(a₀^ε)

∂₁φ₀ ...

∂_nφ₀

⎞

⎟⎟

⎟⎟

⎟⎠

, L= 0 − 0 . . . 0

0 0 . . . 0

0 0 0n×n

, and

A(u, ξ )= n j=1

A_j(u)ξj=

⎛

⎝

v·ξ 0 ^a₂^1tξ 0 v·ξ ^a₂^2tξ 2fa1ξ 2fa2ξ v·ξ In

⎞

⎠,

wherefstands forf(|a1|²+ |a2|²). We now have the system:

∂_tu^ε+ n j=1

A_j(u^ε)∂_ju^ε=ε

2Lu^ε; u^ε_|t=0=u^ε₀. (2.3)

The matricesAj are symmetrized by the matrix S=

I₂ 0 0 _4f¹I_n

,

which is symmetric positive if and only iff(|a1|²+ |a2|²) >0: this includes the case of the decofusing cubic nonlin- earity (1.1), of the cubic-quintic nonlinearity (1.9) withλ >0, and of the cubic-quintic nonlinearity (1.9) withλ <0, provided that|a₁|²+ |a₂|²>|λ|/2.

Proposition 2.1.Assume thatu^ε₀is bounded inX^sfor alls > n/2, uniformly forε∈ [0,1], and that there existsε_∗>0 andδ >0such that

f

|a₀^ε|²

δ >0, ∀x∈Rⁿ, ∀ε∈ [0, ε_∗].

Then fors > n/2+2, there exist T_∗>0 and a unique solution u^ε∈C([0, T∗];X^s)to(2.3)for all ε∈ [0, ε∗]. In addition, this solution is inC([0, T_∗];X^m)for allm > n/2, with bounds independent ofε∈ [0, ε_∗].

Proof. Lets > n/2+2. As usual, the main point consists in obtaininga prioriestimates for the system (2.3), so we shall focus our attention on this aspect. We have ana prioribound foru^εinL^∞:

u^ε(t )

L^∞u^ε₀L^∞+ t 0

n j=1

A_j(u^ε)∂_ju^ε(τ )

L^∞dτ+ t 0

u^ε(τ )

L^∞dτ

u^ε₀L^∞+ t 0

Fu^ε(τ )

L^∞∇u^ε(τ )

H^s−1dτ+C t

0

u^ε(τ )

H^s−2dτ.

We infer:

u^ε(t )

L^∞u^ε₀L^∞+ t 0

Gu^ε(τ )

X^su^ε(τ )

X^sdτ. (2.4)

To have a closed system of estimates, introduceP =(I−)^(s−1)/2∇, so thatfX^s≈fL^∞+ Pf_L². Denote f, g =

Rⁿ

f (x)g(x) dx,

the scalar product inL². SinceSis symmetric, we have d

dt

SPu^ε(t ), Pu^ε(t )

=

∂_tSPu^ε(t ), Pu^ε(t ) +2 Re

S∂_tPu^ε(t ), Pu^ε(t ) . So long as

f

|a^ε|²

δ

2 >0, (2.5)

we have the following set of estimates. First, ∂_tSPu^ε(t ), Pu^ε(t )

∂_tSL^∞Pu^ε(t )²

L²

C_δu^ε(t )

L^∞∂_tu^ε(t )

L^∞u^ε(t )²

X^s. Directly from (2.3), we have:

∂tu^ε(t )

L^∞Cu^ε(t )

L^∞∇u^ε(t )

L^∞+u^ε(t )

L^∞

Cu^ε(t )

X^su^ε(t )

X^s. SinceSLis skew-symmetric, we have

Re

SLPu^ε(t ), Pu^ε(t )

=0,

which prevents any loss of regularity in the estimates. For the quasi-linear term involving the matricesA_j, we note that sinceSA_jis symmetric, commutator estimates (see [20]) yield:

n j=1

SP Aj

u^ε

∂ju^ε

, Pu^ε(t )

Cu^ε(t )

L^∞Pu^ε(t )²

L²∇u^ε(t )

L^∞

Cu^ε(t )

X^sPu^ε(t )²

L². Finally, we have:

d dt

SPu^ε(t ), Pu^ε(t )

Cu^ε(t )

X^su^ε(t )²

X^s.

This estimate, along with (2.4), shows that on a sufficiently small time interval[0, T_∗], withT_∗>0 independent of ε∈ [0, ε_∗], (2.5) holds. This yields the first part of Proposition 2.1.

The fact that the local existence time does not depend ons > n/2+2 follows from the continuation principle based on Moser’s calculus and tame estimates (see e.g. [22, Section 2.2] or [27, Section 16.1]). 2

The existence part of Theorem 1.10 and of Theorem 1.3 whenφ_eik=0 follows. Indeed, defineΦ^εby Φ^ε(t )=φ0−

t 0

1

2v^ε(τ )²+fa^ε(τ )² dτ.

We check that∂_t(∇Φ^ε−v^ε)= ∇∂_tΦ^ε−∂_tv^ε=0, so that∇Φ^ε=v^ε, and(Φ^ε, a^ε)solves (2.2). Finally, uniqueness for (2.1) follows from energy estimates. Ifu^ε, v^ε∈C^∞∩L^∞([0, T_∗] ×Rⁿ)solve (2.1), thenw^ε:=u^ε−v^εsatisfies:

iε∂_tw^ε+ε²

2 w^ε=f

|u^ε|² u^ε−f

|v^ε|²

v^ε; w_|^ε_t=0=0.

We have, fort∈ [0, T_∗], w^εL^∞(0,t;L²)C

u^εL^∞([0,T_∗]×Rⁿ),v^εL^∞([0,T_∗]×Rⁿ)

w^εL¹(0,t;L²), and Gronwall lemma yieldsw^ε≡0.

2.2. Phase-amplitude representation: the caseφ_eik=0

We now consider (1.1)–(1.2) only: the nonlinearity is exactly cubic. To take the presence of V and φ_quad into account, we proceed as in [9]: we construct the solution asu^ε=a^εe^{i(φε+φeik)/ε}. The analogue of (2.2) is:

⎧⎪

⎨

⎪⎩

∂tΦ^ε+1

2|∇Φ^ε|²+V + |a^ε|²−1=0; Φ^ε|t=0=φ0,

∂_ta^ε+ ∇Φ^ε· ∇a^ε+1

2a^εΦ^ε=iε

2a^ε; a^ε|t=0=a₀^ε.

SetΦ^ε=φ^ε+φeik. The introduction ofφeikallows us to get rid of the termsVquad andφquad, and work in Zhidkov spaces. The above problem reads, in terms of(φ^ε, a^ε):

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

∂_tφ^ε+1

2|∇φ^ε|²+ ∇φ_eik· ∇φ^ε+V_lin+ |a^ε|²−1=0,

∂ta^ε+ ∇φ^ε· ∇a^ε+ ∇φeik· ∇a^ε+1

2a^εφ^ε+1

2a^εφeik=iε 2a^ε, φ^ε|t=0=φlin; a^ε|t=0=a^ε₀.

(2.6)

Resume the notations of the previous paragraph, with now:

=

⎛

⎜⎜

⎜⎜

⎝ 0 0

∂₁V_lin ...

∂_nV_lin

⎞

⎟⎟

⎟⎟

⎠, and A(u, ξ )= n j=1

A_j(u)ξj

⎛

⎝

v·ξ 0 ^a₂^1tξ 0 v·ξ ^a₂^2tξ 2a1ξ 2a2ξ v·ξ In

⎞

⎠.

The system (2.3) is replaced by:

∂_tu^ε+ n j=1

A_j u^ε

∂_ju^ε+ ∇φ_eik· ∇u^ε+ ˜Mu^ε+=ε

2Lu^ε; u^ε_|t=0=u^ε₀, (2.7)

whereM˜ = ˜M(t )is a matrix depending on time only, sinceφeikis exactly quadratic inx. This aspect seems necessary in the proof of Proposition 2.2 below. This explains our assumptions, and why we do not content ourselves with general sub-quadratic potential and initial phase as in [9]. The important aspect to notice is that since the nonlinearity in (1.1) isexactlycubic, then the matricesA_j are symmetrized by aconstantmatrix, namely:

S=

I₂ 0 0 ¹₄I_n

.

In [9], nonlinearities which are cubicat the originwere considered (as in [18]), and the possibly quadratic phaseφeik

made the assumptionxa₀^ε∈L²(Rⁿ)apparently necessary, to control the time derivative of the symmetrizer. Of course, we want to avoid this decay assumption for the Gross–Pitaevskii equation, so working with a constant symmetrizer is important.

Proposition 2.2.Assume thatu^ε₀is bounded inX^s for alls > n/2, uniformly forε∈ [0,1]. Then fors > n/2+2, there existT_∗∈ ]0, T], independent ofε∈ [0,1], and a unique solutionu^ε∈C([0, T_∗];X^s)to(2.7). In addition, this solution is inC([0, T∗];X^m)for allm > n/2, with bounds independent ofε∈ [0,1].

Sketch of the proof. The proof follows the same lines as the proof of Proposition 2.1, so we shall only point out the differences.

Let s > n/2+2. By construction, the operator ∂_t + ∇φ_eik· ∇ is a transport operator along the characteristics associated toφ_eik, which do not intersect fort∈ [0, T]. Therefore, we have ana prioribound foru^ε inL^∞:

u^ε(t )

L^∞u^ε₀

L^∞+ t 0

n j=1

Aj(u)∂ju(τ )

L^∞dτ+ t 0

Cu^ε(τ )

L^∞+(τ )

L^∞+u^ε(τ )

L^∞

dτ

u^ε₀

L^∞+C t 0

1+u^ε(τ )

X^su^ε(τ )

X^sdτ+CL^∞([0,T];X^s). (2.8) To have a closed system of estimates, resume the operatorP =(I−)^(s−1)/2∇, so thatfX^s ≈fL^∞+ Pf_L². We have

d dt

SPu^ε(t ), Pu^ε(t )

=2 Re

S∂tPu^ε(t ), Pu^ε(t ) ,

sinceSis constant symmetric. SinceSLis skew-symmetric, we have Re

SLPu^ε(t ), Pu^ε(t )

=0.

For the quasi-linear term involving the matricesA_j, we note that sinceSA_jis symmetric, commutator estimates yield:

n j=1

SP A_j

u^ε

∂_ju^ε

, Pu^ε(t )

Cu^ε(t )

X^sPu^ε(t )²

L². Next, write

SP

∇φ_eik· ∇u^ε(t )

, Pu^ε(t )

=

S∇φ_eik· ∇Pu^ε(t ), Pu^ε(t ) +

S[P ,∇φ_eik· ∇]u^ε(t ), Pu^ε(t ) . The first term of the right-hand side is estimated thanks to an integration by parts:

2 Re

S∇φ_eik· ∇Pu^ε(t ), Pu^ε(t )

=

S∇φ_eik(t, x)· ∇Pu^ε(t, x)²dx

= −

Sφeik(t, x)Pu^ε(t, x)²dx.

For the second term, we notice that[P ,∇φ_eik· ∇] =ψ∇, whereψ=ψ (t, D)is a pseudo-differential operator inx, of orders−1, depending smoothly oft∈ [0, T]. Therefore,

2 Re SP

∇φ_eik· ∇u^ε(t )

, Pu^ε(t )

u^ε(t )²

X^s.

The fact thatM˜ is independent ofx is crucial here, to ensure thatP (Mu˜ ^ε)∈L²foru^ε∈X^s. IfM˜ depended onx, that is ifφ_eikwas not a polynomial of order at most two, the low frequencies might be a problem at this step of the proof. Finally, we have:

d dt

SPu^ε(t ), Pu^ε(t )

Cu^ε(t )

X^s

u^ε(t )²

X^s.

This estimate, along with (2.8), yields the first part of Proposition 2.2. We conclude like in the proof of Proposi- tion 2.1. 2

The existence part of Theorem 1.3 follows from the above result, by setting φ^ε(t )=φlin−

t 0

1

2v^ε(τ )²+ ∇φeik(τ )·v^ε(τ )+Vlin(τ )+a^ε(τ )²−1

dτ.

Finally, uniqueness for (1.1)–(1.2) follows from energy estimates. If u^ε, v^ε ∈C^∞∩L^∞([0, T_∗] ×Rⁿ) solve (1.1)–(1.2), thenw^ε:=u^ε−v^εsatisfies:

iε∂_tw^ε+ε²

2 w^ε=(V −1)w^ε+ |u^ε|²u^ε− |v^ε|²v^ε; w_|^ε_t=0=0.

We have, fort∈ [0, T_∗], w^εL∞(0,t;L²)

u^ε2_L∞([0,T_∗]×Rⁿ)+ v^ε2_L∞([0,T_∗]×Rⁿ))w^εL¹(0,t;L²), and Gronwall lemma yieldsw^ε≡0.

2.3. On the Hamiltonian structure

WhenV =V (x)is time-independent, (1.1) formally has a Hamiltonian structure, with H=1

2ε∇u^ε(t )²

L²+

Rⁿ

V (x)u^ε(t, x)²dx+1

2u^ε(t )²−1²

L².

WhenV ≡0, this structure is used in [17] to prove the global existence of solutions in the energy space. On the other hand, suppose thatV is, say, harmonic:

V (x)= n j=1

λjx_j²,

where the constantsλ_j0 are not all equal to zero. Then necessarily,H is infinite: suppose for instance thatλ₁>0.

Then if∂x₁u^ε(t,·), x1u^ε(t,·)∈L²(Rⁿ), the uncertainty principle (a simple integration by parts, plus Cauchy–Schwarz inequality in this case) yields:

u^ε(t,·)∈L² Rⁿ

.

Therefore, the constraint|u^ε(t,·)|²−1∈L²(Rⁿ)cannot be satisfied, for otherwise, 1=1− |u^ε(t,·)|²+ |u^ε(t,·)|²∈ L²(Rⁿ)+L¹(Rⁿ).

Similarly, assume thatV ≡0, butφ_quad=0: rapid quadratic oscillations are present in the initial data. We have ε∇u^ε_|t=0=

ε∇a₀^ε+ia₀^ε∇φ0

e^iφ0/ε.

Therefore, the above quantity is in L²provided that ∇a^ε₀, a₀^ε∇φ_quad∈L²(Rⁿ). If for instanceφ_quad(x)=cx₁²with c=0, the last assumption means thatx₁a₀^ε∈L²(Rⁿ), which brings us back to the previous discussion.

We shall see in Section 5 that ifφeik≡0, and ifa₀^ε∈X^∞is such that a^ε₀²−1∈L²

Rⁿ ,

then the last constraint present inHis not propagated in general. In small time at least, one has generically u^ε(t,·)²−1∈/L²

Rⁿ . 3. Semi-classical analysis

We now complete the proof of Theorem 1.3. The end of the proof of Theorem 1.10 follows essentially the same lines, so we omit it. The main adaptation is due to the fact that when the nonlinearity is not exactly cubic, the sym- metrizerSis not constant. We refer to [18] or [9], to see that the proof below is easily adapted.

Introduce(φ, a), solution to (2.6) withε=0, that is

⎧⎪

⎨

⎪⎩

∂_tφ+1

2|∇φ|²+ ∇φ_eik· ∇φ+V_lin+ |a|²−1=0; φ|t=0=φ_lin,

∂_ta+ ∇φ· ∇a+ ∇φ_eik· ∇a+1

2aφ+1

2aφ_eik=0; a|t=0=a₀.

(3.1)

It is a particular case of Proposition 2.2 that (3.1) has a unique solution, such that a,∇φ∈C([0, T_∗];X^s)for all s > n/2.

Proposition 3.1.Under the assumptions of Theorem1.3, let(φ^ε, a^ε)and(φ, a)be given by(2.6)and(3.1)respec- tively. For alls > n/2, there existsC_ssuch that

∇(φ^ε−φ)

L^∞([0,T_∗];X^s)+ a^ε−aL^∞([0,T_∗];X^s)C_sε.