Multiphase weakly nonlinear geometric optics for Schrodinger equations
aa r X i v : . [ m a t h . A P ] F e b MULTIPHASE WEAKLY NONLINEAR GEOMETRIC OPTICSFOR SCHR ¨ODINGER EQUATIONS
R´EMI CARLES, ERIC DUMAS, AND CHRISTOF SPARBER
Abstract.
We describe and rigorously justify the nonlinear interaction ofhighly oscillatory waves in nonlinear Schr¨odinger equations, posed on Eu-clidean space or on the torus. Our scaling corresponds to a weakly nonlinearregime where the nonlinearity affects the leading order amplitude of the solu-tion, but does not alter the rapid oscillations. We consider initial states whichare superpositions of slowly modulated plane waves, and use the frameworkof Wiener algebras. A detailed analysis of the corresponding nonlinear wavemixing phenomena is given, including a geometric interpretation on the res-onance structure for cubic nonlinearities. As an application, we recover andextend some instability results for the nonlinear Schr¨odinger equation on thetorus in negative order Sobolev spaces. Introduction
Physical motivation.
The (cubic) nonlinear Schr¨odinger equation (NLS)(1.1) i∂ t u + 12 ∆ u = λ | u | u, with λ ∈ R ∗ , is one of the most important models in nonlinear science. It describesa large number of physical phenomena in nonlinear optics, quantum superfluids,plasma physics or water waves, see e.g. [30] for a general overview. Independentof its physical context one should think of (1.1) as a description of nonlinear wavespropagating in a dispersive medium. In the present work we are interested indescribing the possible resonant interactions of such waves, often referred to as wavemixing . The study of this nonlinear phenomena is of significant mathematical andphysical interest: for example, in the context of fiber optics, where (1.1) describesthe time-evolution of the (complex-valued) electric field amplitude of an opticalpulse, it is known that the dominant nonlinear process limiting the informationcapacity of each individual channel is given by four-wave mixing, cf. [16, 32]. Dueto its cubic nonlinearity, (1.1) seems to be a natural candidate for the investigationof this particular wave mixing phenomena. Similarly, four wave mixing appearsin the context of plasma physics where NLS type models are used to describethe propagation of Alfv´en waves [28]. Moreover, recent physical experiments haveshown the possibility of matter-wave mixing in Bose–Einstein condensates [12]. Aformal theoretical treatment, based on the Gross–Pitaevskii equation ( i.e. a cubicNLS describing the condensate wave function in a mean-field limit), can be foundin [31, 17]. Finally, we also want to mention the closely related studies on so-called
This work was supported by the French ANR project R.A.S. (ANR-08-JCJC-0124-01) andby Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology(KAUST). auto-resonant solutions of NLS given in [13] where again wave mixing phenomenaare used as a method of excitation and control of multi-phase waves.Due to the high complexity of the problem most of the aforementioned works arerestricted to the study of small amplitude waves, representing, in some sense, thelowest order nonlinear effects in systems which can approximately be described by alinear superposition of waves. In addition a slowly varying amplitude approximationis usually deployed. By doing so one restricts himself to resonance phenomenawhich are adiabatically stable over large space- and time-scales. We shall followthis approach by introducing a small parameter 0 < ε ≪
1, which represents themicroscopic/macroscopic scale ratio, and consider a rescaled version of (1.1):(1.2) iε∂ t u ε + ε u ε = λε | u ε | u ε . This is a semi-classically scaled
NLS [6] representing the time evolution of thewave field u ε ( t, x ) on macroscopic length- and time-scales. In the following weseek an asymptotic description of u ε as ε → ε . Note that due to the small parameter ε in front of the nonlinearity,we consider a weakly nonlinear regime . This means that the nonlinearity does notaffect the geometry of the propagation, see § λ (focusing or defocusing nonlinearity) turns out to be irrelevant.1.2. A general formal computation.
In order to describe the appearance ofthe wave mixing in solutions to (1.6), we follow the
Wentzel-Krammers-Brillouin (WKB) approach, as first rigorously settled by Lax [23]. Consider approximatesolutions of (1.2) in the form of high-frequency wave packets, such as(1.3) a ( t, x ) e iφ ( t,x ) /ε . For such a single mode to be an approximate solution, it is necessary that the rapidoscillations are carried by a phase φ which solves the eikonal equation (see [6], wherealso other regimes, in terms of the size of the coupling constant, are discussed):(1.4) ∂ t φ + 12 |∇ φ | = 0 . Nonlinear interactions of high frequency waves are then found by considering su-perpositions of wave packets (1.3). By the cubic interaction, three phases φ , φ and φ generate φ = φ − φ + φ . The corresponding term is relevant at leading order if and only if this new phase φ is characteristic , i.e. solves the eikonal equation (1.4) while also each φ j , j = 1 , , { φ j } j ∈ J , for someindex set J ⊂ Z , such that each φ j is characteristic, and the set is stable under thenonlinear interaction. That is, if k, ℓ, m ∈ J are such that φ = φ k − φ ℓ + φ m ischaracteristic, then φ ∈ { φ j } j ∈ J . Given some index j ∈ J , the set of (four-wave)resonances leading to the phase φ j is then I j = (cid:8) ( k, ℓ, m ) ∈ J ; φ k − φ ℓ + φ m = φ j (cid:9) . One of the tasks of this work to study the structure of I j . A first important step isobtained by plugging φ = φ k − φ ℓ + φ m into (1.4), since then, an easy calculation ULTIPHASE GEOMETRIC OPTICS FOR NLS 3 shows that φ is characteristic if and only if the following resonance condition issatisfied:(1.5) ( ∇ φ ℓ − ∇ φ m ) · ( ∇ φ ℓ − ∇ φ k ) = 0 . Obviously this is a quite severe restriction in one spatial dimension, while in higherdimensions there are many possibilities to satisfy (1.5). In order to gain moreinsight we shall restrict ourselves from now on to the case of plane waves ( i.e. linear phases, see § Basic mathematical setting and outline.
In the following the space vari-able x ∈ M will either belong to the whole Euclidean space M = R d , or to the torus M = T d (we denote T = R / π Z ), for some d ∈ N . The latter can be motivated bythe fact that numerical simulations of (1.6) are mainly based on pseudo-spectralschemes and thus naturally posed on T d , see e.g. [2, 3]. We then consider the initialvalue problem for the slightly more general NLS(1.6) iε∂ t u ε + ε u ε = λε | u ε | σ u ε ; u ε (0 , x ) = u ε ( x ) , where σ ∈ N ∗ . Although we obtain the most precise results (concerning the geom-etry of resonances, in particular) in the case of the cubic nonlinearity ( σ = 1), weare in fact able to rigorously justify WKB asymptotics also for higher order nonlin-earities. We assume that (1.6) is subject to an initial data u ε , which is assumed tobe close (in a sense to be made precise in §
6) to superposition of highly oscillatoryplane waves, i.e. (1.7) u ε ( x ) ≈ X j ∈ J α j ( x ) e iκ j · x/ε , where J ⊆ Z is a (not necessarily finite) given index set. In the Euclidean casewe allow for wave vectors κ j ∈ R d , whereas on M = T d we impose κ j ∈ Z d .Moreover, in the latter case, we choose α j to be independent of x ∈ T d , so that (1.7)corresponds to an expansion in terms of Fourier series (with ε − ∈ N ). The caseof x -dependent α j ’s on T d could be considered as well, by reproducing the analysison R d . We choose not to do so here, since it brings no real new information.In particular, for x ∈ T (the one-dimensional torus), our analysis leads to aremarkably simple approximation. Theorem 1.1.
For x ∈ T , consider (1.6) with σ = 1 . Suppose that the initial dataare of the form (1.7) with κ j ≡ j ∈ Z and ( α j ) j ∈ ℓ ( Z ) .Then for all T > , there exist C = C ( T ) and ε > , such that for all ε ∈ ]0 , ε ] ,with /ε ∈ N ∗ , it holds sup t ∈ [0 ,T ] (cid:13)(cid:13) u ε ( t ) − u ε app ( t ) (cid:13)(cid:13) L ∞ ( T ) Cε, where the approximate solution u ε app is given by u ε app ( t, x ) = X j ∈ Z α j e − iλt (2 M −| α j | ) e i ( jx − j t ) /ε , and M = X k ∈ Z | α k | . R. CARLES, E. DUMAS, AND C. SPARBER
We see that at leading order, the nonlinear interaction shows up through anexplicit modulation at scale O (1). It is well known that the one-dimensional cubicSchr¨odinger equation is completely integrable (see [18, 24] for the periodic case).However, this aspect does not play any role in the proof of Theorem 1.1, which initself can be seen as a consequence of the more general result stated in Theorem 6.5.On the other hand, several aspects in the discussion on possible phase resonancesand the creation of amplitudes seem to be specific to both properties d = 1 and σ = 1 (see § § a j .The second step then consists in making this approach rigorous: we construct theprofiles a j , and show that the obtained ansatz is a satisfactory approximation ofthe exact solution u ε , up to O ( ε ) in a space contained in L ∞ ( M ). As it is standard,we prove in fact a stronger stability result: Starting from any approximate solution u ε app constructed on profiles, we show that, for any initial data close (as ε goes tozero) to u ε app | t=0 , there exists an exact solution which is close to u ε app , on some timeinterval independent of ε (which, for ε small enough, may be chosen as any finitetime up to which u ε app is defined).In the case of a single oscillation only, it suffices to multiply u ε by e − iφ/ε tofilter out rapid oscillations, see [6]. In the case where several phases are present,this strategy obviously fails. To overcome this issue, a fairly general mathematicalapproach, which has proved efficient in several contexts (see e.g. [15, 27, 26]),consists in working in rescaled Sobolev spaces, usually denoted by H sε , for s > ε , in order toaccount for the spatial oscillations at scale ε . More precisely, if s ∈ N , k f k H sε := X | α | s k ( ε∂ ) α f k L . However, due to the negative power of ε in the associated Gagliardo–Nirenberg in-equalities, this technique usually demands to construct approximate solution witha high order of precision (see [14] for a closely related study on the interactionof high-frequency waves in periodic potentials). Another, more sophisticated, ap-proach consists in filtering out the rapid oscillations in terms of the free evolutiongroup, as in [29]. In the present work though, we shall use a simpler approach, whichallows us to justify the multiphase weakly nonlinear WKB analysis in a remark-ably straightforward way. This approach relies on the use of Wiener algebras, asintroduced in [20], and further developed in [22, 4, 9]. This analytical framework isparticularly convenient in the case of plane waves, but could probably be extendedto more general situations, up to some geometric constraints on the phases. How-ever, the first step of the analysis, i.e. describing all possible resonances, becomesmuch more intricate, see e.g. [21, 19].As well shall see during the course of the proof, the use of Wiener algebras hasseveral advantages on the technical level. We point out that this framework makesit possible to justify the WKB approximation with an error estimate of order O ( ε ) ULTIPHASE GEOMETRIC OPTICS FOR NLS 5 without constructing correctors (which would have to be of order ε or even smaller,when working in H sε spaces, see e.g. [15, 27], or [7, 14] in the NLS case).1.4. An application to instability.
As an application of the semi-classical anal-ysis for (1.6), we recover the main result in [8] (see also [5]), concerning NLS inthe periodic case. This result has been established in the case d = 1, and is herebyextend to higher dimensions. We also propose a variation on a result in [25] (seeassertion 3 in the theorem below). Theorem 1.2.
Let d > , σ ∈ N ∗ and λ ∈ {± } . Fix s < . . For all ρ > , we can find a solution u to (1.8) i∂ t u + 12 ∆ u = λ | u | σ u, x ∈ T d , with k u (0) k H s ( T d ) < ρ , such that for all δ > , there exists e u solution to (1.8) with k u (0) − e u (0) k H s ( T d ) < δ, and sup t δ (cid:12)(cid:12)(cid:12)(cid:12)Z T d ( u ( t, x ) − e u ( t, x )) dx (cid:12)(cid:12)(cid:12)(cid:12) > cρ, for some constant c > independent of ρ and δ . In particular, the solution mapfails to be continuous as a map from H s ( T d ) to H k ( T d ) , no matter how close to −∞ the exponent k may be. . Suppose σ > . For any ρ > and δ > there exist smooth solutions u , e u of (1.8) such that u (0) − e u (0) is equal to a constant of magnitude at most δ , and k u (0) k H s ( T d ) + k e u (0) k H s ( T d ) ρ ; sup t δ (cid:12)(cid:12)(cid:12)(cid:12)Z T d ( u ( t, x ) − e u ( t, x )) dx (cid:12)(cid:12)(cid:12)(cid:12) > cρ, for some constant c > independent of ρ and δ . . For any t = 0 , the flow-map associated with (1.8) is discontinuous as a map from L ( T d ) , equipped with its weak topology, into the space of distributions (cid:0) C ∞ ( T d ) (cid:1) ∗ at any constant α ∈ C \ { } ⊂ L ( T d ) . We show in § d = σ = 1, the author showsthe third point in the above statement for any α ∈ L ( T ) \ { } , not necessarilyconstant.1.5. Structure of the paper.
We first study in detail the case of the cubic nonlin-earity ( σ = 1). In §
2, we consider the set of resonant phases, and in §
3, we analyzethe corresponding amplitudes. The case of higher order nonlinearities is treatedin §
4. In §
5, we set up the analytical framework, with which a general stabilityresult (of which Theorem 1.1 is a straightforward consequence) is established in § §
7. Finally, in an appendix, we sketch how the previoussemi-classical analysis can be adapted to more general sets of initial plane waves(including generic finite sets of wave vectors).
Acknowledgments.
The first author wishes to thank Thierry Colin and David Lannesfor preliminary discussions on this subject.
R. CARLES, E. DUMAS, AND C. SPARBER Analysis of possible resonances in the cubic case
In this section, we show that when σ = 1, the set of relevant phases can bedescribed in a fairly detailed way.2.1. General considerations.
We seek an approximation of the form u ε ( t, x ) ≈ X j ∈ J a j ( t, x ) e iφ j ( t,x ) /ε , where here and in the following J ⊂ Z denotes the index set of relevant phases φ j (yet to be determined). Note that using J is only a renumbering, so that j = k ⇒ φ j = φ k . In the case x ∈ T d , one simply drops the dependence of a j upon x . In general J ( J , i.e. we usually need to take into account more phases in (2.1)than we are given initially.As a first step we need to determine the characteristic phases φ j ( t, x ) ∈ R . Forplane-wave initial data of the form (1.7) we are led to the following initial valueproblem ∂ t φ j + 12 |∇ φ j | = 0 ; φ j (0 , x ) = κ j · x, the solution of which is explicitly given by(2.1) φ j ( t, x ) = κ j · x − t | κ j | . Recall that for x ∈ R d , we assume κ j ∈ R d , whereas in the case x ∈ T d , we restrictourselves to κ j ∈ Z d . Of course, these phases φ j remain smooth for all time, i.e.no caustic appears.In the cubic case σ = 1, the set of resonances leading to the phase φ j is thereforegiven by I j = { ( k, ℓ, m ) ∈ J ; κ k − κ ℓ + κ m = κ j , | κ k | − | κ ℓ | + | κ m | = | κ j | } , and the corresponding resonance condition (1.5) becomes(2.2) ( κ ℓ − κ m ) · ( κ ℓ − κ k ) = 0 . As we shall see, this condition provides several insights on the structure of four-waveresonances.2.2.
The one-dimensional case.
For d = 1 the condition (2.2) implies that if( k, ℓ, m ) ∈ I j , then κ ℓ = κ m , or κ ℓ = κ k . Therefore, when d = 1, the set I j is fullydescribed by: I j = { ( j, ℓ, ℓ ) , ( ℓ, ℓ, j ) ; ℓ ∈ J } , no new phase can be generated by a cubic interaction.In higher dimensions, however, the situation is much more complicated andheavily depends on the number of initial modes.2.3. Multi-dimensional case d > . We start with the simplest multiphase situ-ation and proceed from there to more complicated cases. Eventually we shall arriveat a geometric interpretation for the generic case.
ULTIPHASE GEOMETRIC OPTICS FOR NLS 7
One or two initial modes.
If we start from only two initial modes, ♯J = 2,the resonance condition (2.2) implies that the cubic interaction between these twophases cannot create a new characteristic phase. In other words, u ε exhibits atmost two rapid oscillations at leading order. Recalling that φ = 0 is an admissiblephase, the case of a single initial phase ♯J = 1, is therefore included (if one of theinitial amplitudes is set equal to zero). We want to emphasize that the case of atmost two initial phases is rather particular, since (2.2) implies that the situation isthe same for all spatial dimensions d > Remark . In addition, the fact that two phases cannot create a new one extendsalso to higher order (gauge invariant) nonlinearities f ( z ) = λ | z | σ z , for σ ∈ N , σ >
2, see § Three or four initial modes.
This case can be fully understood by the follow-ing geometric insight, already noticed in [10]:
Lemma 2.2.
Let d > , and k, ℓ, m belong to J . Then, ( κ k , κ ℓ , κ m ) ∈ I j pre-cisely when the endpoints of the vectors κ k , κ ℓ , κ m , κ j form four corners of a non-degenerate rectangle with κ ℓ and κ j opposing each other, or when this quadrupletcorresponds to one of the two following degenerate cases: ( κ k = κ j , κ m = κ ℓ ) , or ( κ k = κ ℓ , κ m = κ j ) .Remark . In the degenerate cases, no new phase is created.
Proof.
We recall the argument given in [10], by first noting that the relations be-tween ( κ j , κ k , κ ℓ , κ m ) formulated in (2.1), are equivalently fulfilled by ( κ k − κ, κ ℓ − κ, κ m − κ, κ j − κ ), for any κ ∈ R d (resp. κ ∈ Z d ). This is easily seen by ex-panding the second relation in (2.1) and inserting the first one. Thus, choosing κ = κ j , it therefore suffices to prove this geometric interpretation for κ j = 0, whichconsequently shows: κ k + κ m = κ ℓ such that κ k · κ m = 0, by the law of cosines. (cid:3) In summary, we conclude that three initial (plane-wave) phases create at mostone new phase , such that the corresponding four wave vectors form a rectangle.When the initial wave vectors { κ j } j ∈ J are chosen such that their endpoints formthe four corners of a rectangle, no new phase can be created by the cubic nonlin-earity and u ε exhibits only four rapid oscillations. We close this subsection withtwo illustrative examples. Example . Let d = 2. Consider κ = (0 , κ = (1 ,
1) and κ = (1 , φ ≡ Example . Again let d = 2, with now κ = (1 , κ = (1 ,
2) and κ = (3 , φ , with corresponding wave vector κ = (3 , I j in the general case.2.3.3. The general case.
We are given a countable (possibly finite) number of initialphases { φ j } j ∈ J with corresponding wave vectors { κ j } j ∈ J . From the discussion ofthe previous paragraph it is clear that there are two possible situations:(a) Either, it is impossible to create a new rectangle from any possible subset˜ J ⊂ J , such that ♯ ˜ J = 3. If so, then no new phase can be created. Thisis the generic case. R. CARLES, E. DUMAS, AND C. SPARBER (b) Or, starting from an initial (finite or countable) set S = { κ j } j ∈ J , we mayobtain a first generation S = { κ j } j ∈ J with J ⊂ J ( i.e. S ⊂ S ) inthe following way: we add to S all points κ ∈ R d , such that there exist˜ J ⊂ J with ♯ ˜ J = 3, and such that { κ j } ˜ J ∪ { κ } is a rectangle. Note that,if J ⊂ Z d , then J ⊂ Z d . By a recursive scheme, we are led to a (finiteor countable) set S which is stable under the completion of right-angledtriangles formed of points from this set, into rectangles. Furthermore, if S ⊂ Z d , then S ⊂ Z d . Example . As already seen, the simplest examples for possibility (a) are the cases ♯J ♯J = 4, where the four initial phases are chosen such thattheir corresponding wave vectors { κ j } j ∈ J already form the corners of a rectangle.From a finite number of initial phases, possibility (b) may lead to a finite as wellas to an infinite set J . Even for d = 2, we have: Example . In the plane R , start with J = { ( − , , (0 , , (0 , , (1 , } . The first generation is then J = { ( − , , (0 , , (1 , , ( − , , (0 , , (1 , } = J ∪ { (1 , , ( − , } , and the second one is J = J ∪ { (0 , , (0 , − } . One easily sees that this generates J = Z .As a conclusion, the set of phases { φ j } j ∈ J may be finite or infinite, but has thefollowing property. Proposition 2.8.
Let σ = 1 , and consider any triplet of wave vectors from S = { κ j } j ∈ J . Then, either the corresponding triangle has no right angle, or the fourthcorner of the associated rectangle belongs to S . Analysis of the amplitude system in the cubic case
From the previous section, in general we have to expect the generation of newphases by the four-wave resonance. However, it may happen that not all of them areactually present in our approximation (2.1), since the corresponding profile a j ( t, x )has to be non-trivial.Indeed, if we plug the ansatz (2.1) into (1.6) the terms of order O (1) are iden-tically zero since all the φ j ’s are characteristic. For the O ( ε ) term, we project onthe oscillations associated to φ j , which yields the following system of transportequations:(3.1) ∀ j ∈ J, ∂ t a j + κ j · ∇ a j = − iλ X ( k,ℓ,m ) ∈ I j a k a ℓ a m ; a j (0 , x ) = α j ( x ) , with obviously ∇ a j = 0 in the case where x ∈ T d . In the following we will performa qualitative analysis of the system (3.1), postponing the rigorous existence anduniqueness analysis to § § d = 1 from the case d > ULTIPHASE GEOMETRIC OPTICS FOR NLS 9
The case d = 1 . Let j ∈ J , and recall that I j is particularly simple in d = 1: I j = { ( j, ℓ, ℓ ) , ( ℓ, ℓ, j ) ; ℓ ∈ J } . Using this, (3.1) simplifies to(3.2) ( ∂ t + κ j ∂ x ) a j = − iλ X ℓ ∈ J | a ℓ | a j + iλ | a j | a j ; a j (0 , x ) = α j ( x ) . In particular, the evolution of a zero profile α j ≡ a j ( t, x ) ≡
0. This non-generation of profiles leads to the same conclusion as § § a j factors out in (3.2) just because for any ( ℓ , ℓ , ℓ ) ∈ I j , wehave ℓ = j or ℓ = j ). We shall see that the multi-dimensional situation is quitedifferent but first examine the situation for x ∈ T and x ∈ R in more detail.3.1.1. The case x ∈ T . In this case, we readily obtain that | a j | does not depend ontime. This is due to the fact that (3.2) yields: i∂ t a j ∈ R a j and hence ∂ t | a j | = 0,for all j ∈ Z . In particular we get that M = k u ε (0) k L = X j ∈ J | α j | = k u ε ( t ) k L , ∀ t ∈ R . The conserved quantity M corresponds to the total mass of the exact solution u ε .Using this, we rewrite (3.2) as ddt a j = − iλ (cid:0) M − | α j | (cid:1) a j , which yields an explicit formula for the (global in time) solution a j ( t ) = α j e − iλt ( M −| α j | ) . We observe that in the case of the one-dimensional torus, the interaction of theprofiles a j is particularly simple. Nonlinear effects lead to phase-modulations only.3.1.2. The case x ∈ R . Here, in contrast to the situation on T , the modulus of a j is no longer conserved, since we can only conclude from (3.2) that( ∂ t + κ j ∂ x ) | a j | = 0 , and thus | a j ( t, x ) | = | α j ( x − tκ j ) | . In particular we readily see that for all j ∈ J we have(3.3) k a j ( t ) k L = k α j k L , ∀ t ∈ R . Moreover, we still have an explicit representation for the solution of (3.2) in theform(3.4) a j ( t, x ) = α j ( x − tκ j ) e iS j ( t,x ) , for some real-valued phase S j , yet to be computed. In view of the identity( ∂ t + κ j ∂ x ) a j ( t, x ) = iα j ( x − tκ j ) e iλS j ( t,x ) ( ∂ t + κ j ∂ x ) S j ( t, x ) , equation (3.2) implies(( ∂ t + κ j ∂ x ) S j ( t, x )) α j ( x − tκ j ) = λ − X ℓ ∈ J | α ℓ ( x − tκ ℓ ) | + | α j ( x − tκ j ) | ! × α j ( x − tκ j ) . One easily sees that it is sufficient to impose ∂ t ( S j ( t, x + tκ j )) = − λ X ℓ ∈ J | α ℓ ( x + t ( κ j − κ ℓ )) | + λ | α j ( x ) | , which yields(3.5) S j ( t, x ) = − λ Z t X ℓ ∈ J \{ j } | α ℓ ( x + ( τ − t ) κ j − τ κ ℓ )) | dτ − tλ | α j ( x − tκ j ) | . This formula, together with (3.4) describes the modulation of the profile a j ( t, x ).As in the case of the torus, amplitudes are transported linearly. Only the (slow)phases S j undergo nonlinear effects, which are more complicated as before but stillexplicitly described in terms of the initial data.3.2. The case of one or two modes for d > . We have already seen in § d >
1. Indeed if we start from two phases and two associated profiles, say a j and a ℓ , the system (3.1) simplifies to: ∂ t a j + κ j · ∇ a j = − iλ (cid:0) | a j | + 2 | a ℓ | (cid:1) a j , ; a j (0 , x ) = α j ( x ) ,∂ t a ℓ + κ ℓ · ∇ a ℓ = − iλ (cid:0) | a j | + | a ℓ | (cid:1) a ℓ , ; a ℓ (0 , x ) = α ℓ ( x ) . Note that if initially one of the two profiles is identically zero, it remains zero forall times and hence, we are back in the situation of a usual single-phase WKBapproximation. In particular we compute explicitly for: • Two modes, on T d : a j ( t ) = α j e − iλt (2 | α ℓ | + | α j | ) ; a ℓ ( t ) = α ℓ e − iλt (2 | α j | + | α ℓ | ) . • Two modes, on R d : a j ( t, x ) = α j ( x − tκ j ) e − iλ ( R t | α ℓ ( x +( τ − t ) κ j − τκ ℓ ) | dτ + t | α j ( x − tκ j ) | ) ,a ℓ ( t, x ) = α ℓ ( x − tκ ℓ ) e − iλ ( R t | α j ( x +( τ − t ) κ ℓ − τκ j ) | dτ + t | α ℓ ( x − tκ ℓ ) | ) . Again, these solutions exhibit (nonlinear) self-modulation of phases only, and existfor all times t ∈ R , a property which is a-priori not clear in the general case.3.3. Creation of new modes when d > . A basic difference between the one-dimensional case and the multidimensional situation is that the conservation law(3.3) does not remain valid when d >
2. However, we are still able to prove thatthe total mass is conserved.
Lemma 3.1.
For any solution to (3.1) it holds (3.6) ddt X j ∈ J k a j ( t ) k L = 0 . ULTIPHASE GEOMETRIC OPTICS FOR NLS 11
Proof.
The assertion follows from the more general identity X j ∈ J ( ∂ t + κ j · ∇ ) | a j | = 0 , since, by definition we have X j ∈ J ( ∂ t + κ j · ∇ ) | a j | = Im λ X j ∈ J X ( k,ℓ,m ) ∈ I j a j a k a ℓ a m . This sum is zero by symmetry, since for each quadruplet ( j, k, ℓ, m ) ∈ J of indicesthe quadruplet ( j, m, ℓ, k ) is also present, as well as the other six obtained bycircular permutation (at least in the nondegenerate case mentioned in Lemma 2.2;adaptation to the degenerate case is obvious). These are the only occurrences ofthe corresponding rectangle of wave numbers, and they produce the sum2 ( a j a k a ℓ a m + a k a ℓ a m a j + a ℓ a m a j a k + a m a j a k a ℓ ) = 8 Re ( a j a k a ℓ a m ) , which is real. We consequently infer ∂ t X j ∈ J | a j ( t, x + tκ j ) | = 0 , and thus also ddt X j ∈ J k a j ( t, · + tκ j ) k L = ddt X j ∈ J k a j ( t, · ) k L = 0 . (cid:3) Let us now turn to the possibility of creating new profiles by nonlinear inter-actions (note however that the conservation law (3.6) gives a global constraint onthis process). To simplify the presentation, we assume d = 2. The creation ofnew oscillations in the general case d > R with (0 , . . . , ∈ R d − and analogously for the situation on T d . Consider thegeometry associated to Example 2.4: We thus have (on T d or R d ) i∂ t a = λ X ( k,ℓ,m ) ∈ I a k a ℓ a m . Recall that ( k, ℓ, m ) ∈ I if and only if κ k − κ ℓ + κ m = 0 ; | κ k | − | κ ℓ | + | κ m | = 0 , which obviously implies κ k · κ m = 0. Such a possibility occurs in two cases: • κ k = 0 or κ m = 0. • ( κ k , κ m ) = ( κ , κ ) or ( κ k , κ m ) = ( κ , κ ) and hence κ ℓ = κ .From these various cases, we infer i∂ t a = λ (cid:0) | a | + 2 | a | + | a | + 2 | a | (cid:1) a + 2 a a a . Consider three non-vanishing initial oscillations, such that a a a | t =0 = 0. Thus,even if a | t =0 = 0, we have ∂ t a | t =0 = 0, and this (non-oscillating) fourth mode isinstantaneously non-vanishing. Higher order nonlinearities
Analysis of possible resonances.
So far we were only concerned with four-wave interactions corresponding to cubic nonlinearities, i.e. σ = 1 in (1.6). Ingeneral though, the set of resonances associated to a (gauge invariant) nonlinearityof the form f ( z ) = λ | z | σ z , σ ∈ N , are defined by I σj = n ( ℓ , . . . , ℓ σ +1 ) ∈ J σ +1 ; σ +1 X k =1 ( − k +1 κ ℓ k = κ j , σ +1 X k =1 ( − k +1 | κ ℓ k | = | κ j | o . As in Section 2, the set of wave vectors { κ j } j ∈ J is constructed by induction, startingfrom an a finite or countable set { κ j } j ∈ J , to which we first add a vector κ whenthere exist κ ℓ , . . . , κ ℓ σ +1 ∈ J such that(4.1) σ +1 X k =1 ( − k +1 | κ ℓ k | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ +1 X k =1 ( − k +1 κ ℓ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ;we then set κ = P σ +1 k =1 ( − k +1 κ ℓ k . The same iterative procedure as in § Proposition 4.1.
Let σ > , and consider any (2 σ + 1) -tuple ( κ ℓ , . . . , κ ℓ σ +1 ) ofwave vectors from S = { κ j } j ∈ J . Then, either the relation (4.1) is not satisfied, orthe vector κ j = P σ +1 k =1 ( − k +1 κ ℓ k belongs to S .Remark . It is worth noting that, even if we only have very poor informationon the set of wave vectors { κ j } j ∈ J , it is however a subset of the group generatedby the initial set { κ j } j ∈ J .The profile equations, analogue to (3.1), are then, for all j ∈ J :(4.2) ∂ t a j + κ j · ∇ a j = − iλ X ( ℓ ,...,ℓ σ +1 ) ∈ I j a ℓ a ℓ . . . a ℓ σ +1 ; a j (0 , x ) = α j ( x ) . The case of two modes.
Similar to the situation for σ = 1, the case ofonly two initial modes is rather special. Indeed, the fact that two phases cannotcreate a new one extends also to higher order nonlinearities. In order to explain theargument, consider first a quintic nonlinearity, corresponding to σ = 2. To obtaina nonlinear resonance, the wave vectors need to satisfy κ k − κ ℓ + κ m − κ p + κ q = κ j , | κ k | − | κ ℓ | + | κ m | − | κ p | + | κ q | = | κ j | , where k, ℓ, m, p, q ∈ { j , j } , j , j ∈ J . First, if j (or j ) appears at least twice onthe left hand side, with at least one plus and one minus, then the cancellation re-duces the discussion to the one we had about the cubic nonlinearity. Hence, no newresonant phase can be created in this case. The complementary case corresponds,up to exchanging j and j , to κ k = κ m = κ q = κ j and κ ℓ = κ p = κ j . The above relations yield3 κ j − κ j = κ j ; 3 | κ j | − | κ j | = | κ j | . Squaring the first identity and comparing with the second one, we infer6 | κ j − κ j | = 0 . ULTIPHASE GEOMETRIC OPTICS FOR NLS 13
Therefore, no new resonant phase can be created by the quintic interaction of twoinitial resonant plane waves.Consider now the general case where σ >
2: The same argument as above showsthat the only new case is the one where all the plus signs correspond to one phase,and all the minus signs to the other:( σ + 1) κ j − σκ j = κ j ; ( σ + 1) | κ j | − σ | κ j | = | κ j | . Squaring the first identity and comparing with the second one, we infer σ ( σ + 1) | κ j − κ j | = 0 . We conclude as above, and obtain the following result:
Proposition 4.3.
Let σ ∈ N ∗ , and let κ , κ ∈ R d be such that κ = κ . To thesewave vectors are associated the characteristic phases φ j ( t, x ) = κ j · x − t | κ j | , j = 1 , . Then, these two phases can not create no new phase by (2 σ +1) th-order interaction:the set n κ ∈ R d | ∃ ( ℓ , . . . , ℓ σ +1 ) ∈ { , } σ +1 , κ = σ +1 X k =1 ( − k +1 κ ℓ k and | κ | = σ +1 X k =1 ( − k +1 | κ ℓ k | ) o is reduced to { κ , κ } . In view of Proposition 4.3, the system (4.2) becomes a system of two equations,which can be integrated explicitly, as in [8, Remark 3.1]:(4.3) ∂ t a j + κ j · ∇ a j = − iλ σ X n =0 (cid:18) σ + 1 n (cid:19) (cid:18) σn (cid:19) | a j | σ − n | a ℓ | n a j ,∂ t a ℓ + κ ℓ · ∇ a ℓ = − iλ σ X n =0 (cid:18) σ + 1 n (cid:19) (cid:18) σn (cid:19) | a ℓ | σ − n | a j | n a ℓ . In the case of T d , we find for instance(4.4) a j ( t ) = α j exp − iλt σ X n =0 (cid:18) σ + 1 n (cid:19) (cid:18) σn (cid:19) | α j | σ − n | α ℓ | ℓ ! ,a ℓ ( t ) = α ℓ exp − iλt σ X n =0 (cid:18) σ + 1 n (cid:19) (cid:18) σn (cid:19) | α ℓ | σ − n | α j | n ! . In the case of R d , the formula is more intricate and we shall omit it.Apart from the two-phase situation, the results for of Section 2.3.3 on resonancesdo not carry over to the general case σ > d = 1, the resonant sets cease to be as simple for σ >
2, providedthat one starts with at least three modes.
Example . Consider the quintic case σ = 2 in d = 2 spatial dimensions. As wehave seen above a resonance for such a quintic nonlinearity appears if and only if κ k − κ ℓ + κ m − κ p + κ q = κ j , | κ k | − | κ ℓ | + | κ m | − | κ p | + | κ q | = | κ j | . We can pick for instance three initial phases of the form κ = ( − ,
0) ; κ = (0 ,
0) ; κ = (2 , . For k = 1, ℓ = p = 2, m = q = 3, we have a resonance, creating κ = (3 , σ = 1, no resonance occurs between the phases with wave vectors κ , κ and κ . This example shows that the geometric characterization of four-waveresonances given in § κ , κ , κ and κ all belong to the line x = 0. Example . Consider the same example as above in d = 1. i.e. pick three initialphases of the form κ = − κ = 0 ; κ = 2and create a resonance κ = 3 for k = 1, ℓ = p = 2, m = q = 3. This is in sharpcontrast to the case σ = 1, where no new phases can be created in d = 1. Moreover,a non-vanishing amplitude a is effectively generated:( ∂ t + κ ∂ x ) a = − iλ (cid:0) | a | + | a | + | a | + 4( | a | | a | + | a | | a | + | a | | a | ) (cid:1) a − i λa a a − iλa | a | a . We see that we may have a | t =0 = 0, but( ∂ t a ) | t =0 = (cid:0) − i λa a a − iλa | a | a (cid:1) | t =0 = 0 , showing the appearance of a non-trivial a for t > Analytical framework
We now present the analytical framework needed for the rigorous justification ofa multiphase WKB approximation.5.1.
Wiener algebras. On M = T d , we consider the usual Wiener algebra offunctions with absolutely summable Fourier series: Definition 5.1 (Wiener algebra on M = T d ) . Functions of the form f ( y ) = X k ∈ Z b k e iκ k · y , with κ k ∈ Z d and b k ∈ C , belong to W ( T d ) if and only if ( b k ) k ∈ Z ∈ ℓ ( Z ) . We denote k f k W = X k ∈ Z | b k | . In the sequel, when x ∈ T d , we consider initial data for (1.6) which are of theform f ( x/ε ), with f ∈ W ( T d ) and ε − ∈ N ∗ . ULTIPHASE GEOMETRIC OPTICS FOR NLS 15
Lemma 5.2.
Let f belong to W ( T d ) . Then, for all ε > such that ε − ∈ N ∗ , wehave f ( · /ε ) ∈ W ( T d ) , and k f ( · /ε ) k W = k f k W . For M = R d , the framework is a bit different. Define the Fourier transform by F f ( ξ ) = b f ( ξ ) = 1(2 π ) d/ Z R d f ( x ) e − ix · ξ dx. With this normalization, we have F − f ( x ) = F f ( − x ). Following [20] and [9], weuse on R d two different Wiener-type algebras: For the exact solution we use W ( R d ), i.e. the space of functions with Fourier transform in L ( R d ), and for the profiles,we use A ( R d ), the space of almost periodic W ( R d )-valued functions on R d , withabsolutely summable Fourier series. We also set A ( T d ) = W ( T d ), equipped withthe same norm. Definition 5.3 (Wiener algebra on M = R d ) . We define W ( R d ) = n f ∈ S ′ ( R d ; C ) , k f k W := k b f k L ( R d ) < ∞ o . Functions of the form f ( x, y ) = X k ∈ Z b k ( x ) e iκ k · y , with κ k ∈ R d and b k ∈ W ( R d ) , belong to A ( R d ) if and only if k f k A := X k ∈ Z k b k k W = X k ∈ Z k b b k k L ( R d ) < ∞ . In the sequel, when x ∈ R d , we consider initial data for (1.6) which are of theform f ( x, x/ε ), with f ∈ A ( R d ). Again, we have Lemma 5.4.
Let f ∈ A ( R d ) and ε > . Then f ( · , · /ε ) ∈ W ( R d ) and k f ( · , · /ε ) k W k f k A . Proof.
We simply have, when f ( x, y ) = P k ∈ Z b k ( x ) e iκ k · y : k f ( · , · /ε ) k W = (cid:13)(cid:13) X k ∈ Z b b k ( ·− κ k /ε ) (cid:13)(cid:13) L ( R d ) X k ∈ Z k b b k ( ·− κ k /ε ) k L ( R d ) = X k ∈ Z k b b k k L ( R d ) . The last term is, by definition, k f k A . (cid:3) Denote (in the periodic setting as well as in the Euclidean case) U ε ( t ) = e iε t ∆ . The following properties will be useful (see [9], and also [20, 22, 4]).
Lemma 5.5.
Let M = T d or R d . . W ( M ) is a Banach space, continuously embedded into L ∞ ( M ) . . W ( M ) is an algebra, in the sense that the mapping ( f, g ) f g is continuousfrom W ( M ) to W ( M ) , and moreover ∀ f, g ∈ W ( M ) , k f g k W k f k W k g k W . . If F : C → C maps u to a finite sum of terms of the form u p u q , p, q ∈ N , then itextends to a map from W ( M ) to itself which is uniformly Lipschitzean on boundedsets of W ( M ) . . For all t ∈ R , U ε ( t ) is unitary on W ( M ) . Action of the free Schr¨odinger group on W ( M ) . As it is standard forsolutions to the equation iε∂ t w ε + ε w ε = F ε , we will consider the corresponding Duhamel’s formula w ε ( t, x ) = U ε ( t ) w ε (0 , x ) − iε − Z t U ε ( t − τ ) F ε ( τ, x ) dτ. In view of this representation formula we first need to study the action of thefree Schr¨odinger group U ε ( t ) on W ( M ).5.2.1. The case M = T d . The action of U ε ( t ) on Fourier series on T d is wellunderstood. For P k ∈ Z b k e iκ k · y ∈ W ( T d ):(5.1) U ε ( t ) X k ∈ Z b k e iκ k · x/ε ! = X k ∈ Z b k e iκ k · x/ε − i | κ k | t/ (2 ε ) . In view of Duhamel’s formula, we will use the following
Lemma 5.6.
Let
T > , ω ∈ Z , κ ∈ Z d , and b, ∂ t b ∈ L ∞ ([0 , T ]) . Denote D ε ( t, x ) := Z t U ε ( t − τ ) (cid:16) b ( τ ) e iκ · x/ε − iωτ/ (2 ε ) (cid:17) dτ. . We have D ε ∈ C ([0 , T ] × T d ) and k D ε k L ∞ ([0 ,T ] × T d ) Z T | b ( t ) | dt. . Assume ω = | κ | . Then there exists C independent of κ , ω and b such that k D ε k L ∞ ([0 ,T ] × T d ) Cε || κ | − ω | (cid:0) k b k L ∞ ([0 ,T ]) + k ∂ t b k L ∞ ([0 ,T ]) (cid:1) . Proof.
In view of the identity (5.1), we have D ε ( t, x ) = Z t b ( τ ) e iκ · x/ε − iωτ/ (2 ε ) e − i | κ | ( t − τ ) / (2 ε ) dτ = e iκ · x/ε − i | κ | t/ (2 ε ) Z t b ( τ ) e i ( | κ | − ω ) τ/ ε dτ. The first point is straightforward. Integration by parts yields, since by assumption | κ | − ω ∈ Z \ { } : with φ ( t, x ) = κ · x − | κ | t/ D ε ( t, x ) = e iφ ( t,x ) /ε (cid:16) − εi | κ | − ω b ( τ ) e i ( | κ | − ω ) τ/ ε (cid:12)(cid:12)(cid:12) t + 2 εi | κ | − ω Z t ∂ t b ( τ ) e i ( | κ | − ω ) τ/ ε dτ (cid:17) . The lemma then follows easily. (cid:3)
ULTIPHASE GEOMETRIC OPTICS FOR NLS 17
The case M = R d . The Euclidean counterpart of Lemma 5.6 is a little bitmore delicate:
Lemma 5.7.
Let
T > , ω ∈ R , κ ∈ R d , and b ∈ L ∞ ([0 , T ]; W ( R d )) . Denote D ε ( t, x ) := Z t U ε ( t − τ ) (cid:16) b ( τ, x ) e iκ · x/ε − iωτ/ (2 ε ) (cid:17) dτ. . We have D ε ∈ C ([0 , T ]; W ( R d )) and k D ε k L ∞ ([0 ,T ]; W ) Z T k b ( t, · ) k W dt. . Assume ω = | κ | , and ∂ t b, ∆ b ∈ L ∞ ([0 , T ]; W ) . Then we have the control k D ε k L ∞ ([0 ,T ]; W ) Cε || κ | − ω | (cid:16) k b k L ∞ ([0 ,T ]; W ) + k ∆ b k L ∞ ([0 ,T ]; W ) + k ∂ t b k L ∞ ([0 ,T ]; W ) (cid:17) , where C is independent of κ , ω and b .Proof. By the definition of U ε ( t ), we have b D ε ( t, ξ ) = Z t e − iε ( t − τ ) | ξ | / b b (cid:16) τ, ξ − κε (cid:17) e − iωτ/ (2 ε ) dτ. Setting η = ξ − κ/ε , we have b D ε ( t, ξ ) = e − iεt | η + κ/ε | / Z t e iετ | η + κ/ε | / b b ( τ, η ) e − iωτ/ (2 ε ) dτ = e − iεt | η + κ/ε | / Z t e iτθ/ b b ( τ, η ) dτ, where we have denoted θ = ε (cid:12)(cid:12)(cid:12) η + κε (cid:12)(cid:12)(cid:12) − ωε = ε | η | + 2 κ · η | {z } θ + | κ | − ωε | {z } θ . The first point of the lemma is straightforward. To prove the second point, integrateby parts, by first integrating e iτθ / : b D ε ( t, ξ ) = − iθ e iτθ/ b b ( τ, η ) (cid:12)(cid:12)(cid:12) t + 2 iθ Z t e iτθ/ (cid:18) i θ b b ( τ, η ) + c ∂ t b ( τ, η ) (cid:19) dτ. We infer, if b, ∂ t b, ∆ b ∈ L ∞ ([0 , T ]; W ):sup t ∈ [0 ,T ] k b D ε ( t ) k L . | θ | (cid:16) k b b k L ∞ ([0 ,T ]; L ) + k c ∆ b k L ∞ ([0 ,T ]; L ) + k c ∂ t b k L ∞ ([0 ,T ]; L ) (cid:17) . This yields the second point of the lemma. (cid:3)
Construction of the exact solution.
As a preliminary step in establishinga WKB approximation we first need to know that (1.6) is well posed on W ( M ). Proposition 5.8.
Consider the initial value problem (5.2) iε∂ t u ε + ε u ε = λε | u ε | σ u ε ; u ε (0 , x ) = u ε ( x ) , where σ ∈ N ∗ , λ ∈ R , and x ∈ M , with either M = R d , or M = T d , in whichcase ε − ∈ N ∗ . If u ε ∈ W ( M ) , then there exists T ε > and a unique solution u ε ∈ C ([0 , T ε ]; W ( M )) to (5.2) . Remark . At this stage, the dependence of T ε upon ε is unknown. In particular, T ε might go to zero as ε →
0. The proof below actually shows that if u ε is uniformlybounded in W ( M ) for ε ∈ ]0 , T ε > ε . Thiscase includes initial data (1.7) which we consider for the WKB analysis. Proof.
Duhamel’s formulation of (5.2) reads u ε ( t ) = U ε ( t ) u ε − iλ Z t U ε ( t − τ ) (cid:0) | u ε | σ u ε ( τ ) (cid:1) dτ. Denote by Φ ε ( u ε )( t ) the right hand side in the above formula. From Lemmae 5.5,5.6 and 5.7, we have: k Φ ε ( u ε )( t ) k W k u ε k W + | λ | Z t k u ε ( τ ) k σ +1 W dτ, and if k u ε k L ∞ ([0 ,T ]; W ) , k e u ε k L ∞ ([0 ,T ]; W ) R , then there exists C = C ( R ) such that k Φ ε ( u ε )( t ) − Φ ε ( e u ε )( t ) k W C ( R ) Z t k u ε ( τ ) − e u ε ( τ ) k W dτ, ∀ t ∈ [0 , T ] . A fixed point argument in ( u ∈ C ([0 , T ]; W ( M )) , sup t ∈ [0 ,T ] k u ( t ) k W k u ε k W ) for T = T ε > (cid:3) Construction of the profiles.
In order to justify our multiphase WKB anal-ysis, we first need to establish an existence theory for the system of profile equations.To this end, for all σ ∈ N ∗ , we rewrite the system (4.2) in its integral form:(5.3) ∀ j ∈ J, a j ( t, x ) = a j (0 , x − tκ j ) − iλ Z t N σ ( a, . . . , a )( τ, x + ( τ − t ) κ j ) dτ, where, for a (1) = ( a (1) j ) j ∈ J , . . . , a (2 σ +1) = ( a (2 σ +1) j ) j ∈ J , we define the nonlinearterm N σ by: ∀ j ∈ J, N σ (cid:16) a (1) , . . . , a (2 σ +1) (cid:17) = X ( ℓ ,...,ℓ σ +1 ) ∈ I j a (1) ℓ a (2) ℓ . . . a (2 σ ) ℓ σ a (2 σ +1) ℓ σ +1 . It is clearly linear with respect to its arguments with odd exponents, and anti-linearwith respect to the others. We prove in Lemma 5.11 below that it is in fact welldefined and continuous on E ( M ), for M = T d or M = R d : Definition 5.10.
Define E ( R d ) = { a = ( a j ) j ∈ J | ( b a j ) j ∈ J ∈ ℓ ( J ; L ( R d )) } , equipped with the norm k a k E ( R d ) = X j ∈ J k b a j k L . Set also E ( T d ) = ℓ ( J ) , equipped with the usual norm k a k E ( T d ) = X j ∈ J | a j | . ULTIPHASE GEOMETRIC OPTICS FOR NLS 19
Note that E simply represents, via an isometric correspondence, the family ofcoefficients of functions in A (up to the choice of the wave numbers κ j in the caseof R d ): f ( x, y ) = X j ∈ J a j ( x ) e iκ j · y ∈ A ( R d ) iff a ∈ E ( R d ) , and then k a k E = k f k A . The same holds for M = T d . Lemma 5.11.
Let σ ∈ N ∗ . For M = R d or M = T d , the nonlinear expres-sion N σ defines a continuous mapping from E ( M ) σ +1 to E ( M ) , and for all a (1) , . . . , a (2 σ +1) ∈ E ( M ) (cid:13)(cid:13)(cid:13) N σ (cid:16) a (1) , . . . , a (2 σ +1) (cid:17)(cid:13)(cid:13)(cid:13) E k a (1) k E . . . k a (2 σ +1) k E . Proof.
We consider the case M = R d , since M = T d is even simpler. In order tobound (cid:13)(cid:13)(cid:13) N σ (cid:16) a (1) , . . . , a (2 σ +1) (cid:17)(cid:13)(cid:13)(cid:13) E == X j ∈ J (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X ( ℓ ,...,ℓ σ +1 ) ∈ I j F (cid:16) a (1) ℓ (cid:17) ∗ F (cid:16) a (2) ℓ (cid:17) ∗ · · · ∗ F (cid:16) a (2 σ +1) ℓ σ +1 (cid:17) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L X j ∈ J X ( ℓ ,...,ℓ σ +1 ) ∈ I j (cid:13)(cid:13)(cid:13) F (cid:16) a (1) ℓ (cid:17) ∗ F (cid:16) a (2) ℓ (cid:17) ∗ · · · ∗ F (cid:16) a (2 σ +1) ℓ σ +1 (cid:17)(cid:13)(cid:13)(cid:13) L , we use Young’s inequality and observe that, once j , ℓ , . . . , ℓ σ are chosen, ℓ σ +1 is determined (since κ ℓ σ +1 = κ j − P σk =1 ( − k +1 κ ℓ k , and n = m ⇒ κ n = κ m ), sothat (cid:13)(cid:13)(cid:13) N σ (cid:16) a (1) , . . . , a (2 σ +1) (cid:17)(cid:13)(cid:13)(cid:13) E X ( ℓ ,...,ℓ σ +1 ) ∈ J σ +1 (cid:13)(cid:13)(cid:13) F (cid:16) a (1) ℓ (cid:17)(cid:13)(cid:13)(cid:13) L . . . (cid:13)(cid:13)(cid:13) F (cid:16) a (2 σ +1) ℓ σ +1 (cid:17)(cid:13)(cid:13)(cid:13) L , which gives the desired result. (cid:3) This consequently yields the following existence result for (5.3), where here andin the following we denote h κ i ≡ | κ | . Proposition 5.12.
Let σ ∈ N ∗ , and M = R d or M = T d .For all α = ( α j ) j ∈ J ∈ E ( M ) , there exist T > and a unique solution t a ( t ) = ( a j ( t )) j ∈ J ∈ C ([0 , T ] , E ( M )) to the system (5.3) , with a (0) = α . Moreover, the following properties hold: . If ( h κ j i s α j ) j ∈ J ∈ E ( M ) for some s ∈ N , then ( h κ j i s a j ) j ∈ J ∈ C ([0 , T ] , E ( M )) . . On M = R d , if ( h κ j i s ∂ βx α j ) j ∈ J ∈ E ( R d ) , for some β ∈ N d and s ∈ N , then ( h κ j i s ∂ βx a j ) j ∈ J ∈ C ([0 , T ]; E ( R d )) .Proof. The existence result follows from Lemma 5.11 and the standard Cauchy–Lipschitz result for ODE’s. Concerning the propagation of moments h κ j i s a j , weagain apply a fixed-point argument, estimating nonlinear terms h κ j i s N σ ( a (1) , . . . , a (2 σ +1) ) as in the proof of Lemma 5.11, via h κ j i ≡ | κ j | = 1 + σ +1 X k =1 ( − k +1 | κ ℓ k | σ +1 X k =1 h κ ℓ k i (2 σ + 1) σ +1 Y k =1 h κ ℓ k i , when ( ℓ , . . . , ℓ σ +1 ) ∈ I j . The last statement of the proposition is concerned withthe smooth dependence upon the parameter x . This follows by commuting (4.2)with ∂ x and using the fact that W ( R d ) is an algebra, continuously embedded in L ∞ , since then ddt k ∂ x a k E . k ∂ x α k E + C ( k a k E ) k ∂ x a k E , and a Gronwall argument shows that k ∂ x a k E remains bounded for all t ∈ [0 , T ].Similarly we conclude for the higher order derivatives, possibly multiplied by weights h κ j i s . (cid:3) For the particular situation for σ = 1, in d = 1 and/or the case of only two initialphases, we infer a stronger result, thanks to the explicit formulas given in § § Corollary 5.13.
Under the assumption of Proposition 5.12, in the case σ = 1 , ifin addition d = 1 , then T can be taken arbitrarily large, with a j ( t ) explicitly givenby (3.4) and (3.5) . Similarly, if ♯J , then T can be taken arbitrarily large.Remark . In the case of higher order nonlinearities, i.e. σ >
2, Equation (4.3)makes it possible to see, via explicit integration (see (4.4) in the case of the torus),that if α j , α ℓ ∈ W ( M ), then a j , a ℓ ∈ C ([0 , ∞ [ , W ( M )).6. Rigorous justification of the multiphase WKB analysis
Construction of an approximate solution.
We start from oscillating ini-tial data, given by a profile in A ( M ), with M = T d or R d : u ε app (0 , x ) = X j ∈ J α j ( x ) e iκ j · x/ε , with α j ( x ) = Const . in the case M = T d . Assumption 6.1.
For both M = R d and M = T d we assume ( α j ) j ∈ J ∈ E ( M ) .For M = R d we assume in addition ∀| β | , ( ∂ βx α j ) j ∈ J ∈ E ( R d ) , and ∀| β | , ( h κ j i ∂ βx α j ) j ∈ J ∈ E ( R d ) . From Proposition 5.12 we know, that these data produce a solution ( a j ) j ∈ J ∈ C ([0 , T ] , E ( M )) to the amplitude system and we consequently define the approxi-mate solution u ε app by(6.1) u ε app ( t, x ) = X j ∈ J a εj ( t, x ) e iφ j ( t,x ) /ε , with φ j given by (2.1). The sequence ( a j ) j ∈ J is such that (cid:0) ∂ βx a j (cid:1) j ∈ J ∈ C ([0 , T ] , E ( M )) , | β | , (cid:0) h κ j i ∂ βx a j (cid:1) j ∈ J ∈ C ([0 , T ] , E ( M )) , | β | . ULTIPHASE GEOMETRIC OPTICS FOR NLS 21
We see from equation (5.3) that ( ∂ t a j ) j ∈ J ∈ C ([0 , T ] , E ( M )). We find (in the senseof distributions) iε∂ t u ε app + ε u ε app = λε | u ε app | σ u ε app − λεr ε + ε r ε , where(6.2) r ε = 12 X j ∈ J e iφ j /ε ∆ a j , and the remainder r ε takes into account the non-characteristic phases created bynonlinear interaction. This means that it is a sum of terms of the form a ℓ a ℓ . . . a ℓ σ a ℓ σ +1 e i ( φ ℓ − φ ℓ + ... − φ ℓ σ + φ ℓ σ +1 ) /ε , where the rapid phase is given by σ +1 X p =1 ( − p +1 φ ℓ p ( t, x ) = σ +1 X p =1 ( − p +1 κ ℓ p ! · x − t σ +1 X p =1 ( − p +1 | κ ℓ p | , and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ +1 X p =1 ( − p +1 κ ℓ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = σ +1 X p =1 ( − p +1 (cid:12)(cid:12) κ ℓ p (cid:12)(cid:12) . In other words, ( ℓ , . . . , ℓ σ +1 ) belongs to the non-resonant set N := J σ +1 \ [ j ∈ J I σj . With these conventions, we have(6.3) r ε = X ( ℓ ,...,ℓ σ +1 ) ∈ N a ℓ a ℓ . . . a ℓ σ +1 e i ( φ ℓ − φ ℓ + ... − φ ℓ σ + φ ℓ σ +1 ) /ε . Estimating r ε in W is straightforward, since ( ∂ βx a j ) j ∈ J ∈ C ([0 , T ] , E ) for | β | k r ε k W k ∆ a k E . Note that r simply vanishes if M = T d . In order to estimate r ε , we impose thefollowing condition on the set of wave numbers { κ j } j ∈ J . Assumption 6.2.
There exists c > such that for all ( ℓ , . . . , ℓ σ +1 ) ∈ N , δ ( ℓ , . . . , ℓ σ +1 ) ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ +1 X p =1 ( − p +1 κ ℓ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − σ +1 X p =1 ( − p +1 (cid:12)(cid:12) κ ℓ p (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > c. Remark . ( i ) This assumption is of course satisfied when only finitely manyphases are created ♯J < ∞ .( ii ) Similarly, this assumption holds for { κ j } j ∈ J ⊂ Z d , since in this case, thequantity considered is an integer.( iii ) Consider the cubic case σ = 1, and suppose that { κ j } j ∈ J is included in arectangular net. Up to translation, this rectangular net has the form { Am ∈ R d | m ∈ Z d } , with A a d × d matrix of the form A = RD , where D is diagonal, and R is a rotation.Then we have, for all k, l, m ∈ Z d : (cid:12)(cid:12)(cid:12) | Ak − Aℓ + Am | − | Ak | + | Aℓ | − | Am | (cid:12)(cid:12)(cid:12) = | ( Ak − Aℓ ) · ( Ak − Am ) | = (cid:12)(cid:12) ( k − l ) · (cid:0) ( A T A )( k − m ) (cid:1)(cid:12)(cid:12) . Since T AA = D , denoting µ , . . . , µ d the squares of the eigenvalues of D , Assump-tion 6.2 is then satisfied if and only if the group generated by µ , . . . , µ d in R isdiscrete, i.e. these numbers are (pairwise) rationally dependent.The reason for imposing the above assumption is a small divisor problem, aswill become clear from the proof of the following lemma. It is possible to relaxAssumption 6.2 to a less rigid one, to the cost of a more technical presentation.The latter is sketched in an appendix. Lemma 6.4.
For M = T d or M = R d , let r ε be given by (6.3) and denote (6.5) R ε ( t, x ) := Z t U ε ( t − τ ) r ε ( τ, x ) dτ, on [0 , T ] × M . Let Assumptions 6.1–6.2 hold. Then, there exists a constant
C > , such that: k R ε k L ∞ ([0 ,T ]; W ( M )) Cε.
Proof.
We only treat the case on M = R d in detail. The case M = T d can betreated analogously. We have R ε ( t, x ) = X ( ℓ ,...,ℓ σ +1 ) ∈ N Z t U ε ( t − τ ) (cid:16) ( a ℓ a ℓ . . . a ℓ σ +1 ) e i ( φ ℓ − φ ℓ + ··· + φ ℓ σ +1 ) /ε (cid:17) ( τ, x ) dτ. Thus, setting b ℓ ,...,ℓ σ +1 := a ℓ a ℓ . . . a ℓ σ +1 , Lemma 5.7 yields k R ε k L ∞ ([0 ,T ]; W ) . ε X ( ℓ ,...,ℓ σ +1 ) ∈ N δ ( ℓ , . . . , ℓ σ +1 ) (cid:16) k b b ℓ ,...,ℓ σ +1 k L ∞ ([0 ,T ]; L ) + k c ∆ b ℓ ,...,ℓ σ +1 k L ∞ ([0 ,T ]; L ) + k c ∂ t b ℓ ,...,ℓ σ +1 k L ∞ ([0 ,T ]; L ) (cid:17) . ε X ( ℓ ,...,ℓ σ +1 ) ∈ N (cid:16) k b b ℓ ,...,ℓ σ +1 k L ∞ ([0 ,T ]; L ) + k c ∆ b ℓ ,...,ℓ σ +1 k L ∞ ([0 ,T ]; L ) + k c ∂ t b ℓ ,...,ℓ σ +1 k L ∞ ([0 ,T ]; L ) (cid:17) , where we have used Assumption 6.2. Next, using Young’s inequality, as in the proofof Lemma 5.11, we get: X ( ℓ ,...,ℓ σ +1 ) ∈ N k b b ℓ ,...,ℓ σ +1 k L ∞ ([0 ,T ]; L ) . X ( ℓ ,...,ℓ σ +1 ) ∈ N k b a ℓ k L ∞ T L . . . k b a ℓ σ +1 k L ∞ T L . X ( ℓ ,...,ℓ σ +1 ) ∈ J σ +1 k b a ℓ k L ∞ T L . . . k b a ℓ σ +1 k L ∞ T L . k ( a j ) j ∈ J k σ +1 L ∞ ([0 ,T ]; E ) . Leibniz formula and H¨older inequality yield similar estimates for c ∆ b ℓ ,...,ℓ σ +1 and c ∂ t b ℓ ,...,ℓ σ +1 in L ∞ ([0 , T ]; L ( M )), and the lemma follows. (cid:3) ULTIPHASE GEOMETRIC OPTICS FOR NLS 23
Accuracy of the multiphase WKB approximation.
With the above re-sults in hand, we can now prove our main theorem.
Theorem 6.5 (General approximation result) . Let σ > , M = T d or R d , andAssumptions 6.1–6.2 hold. Given an approximate solution u ε app ∈ C ([0 , T ]; W ( M )) as in (6.1) , we consider a family of initial data ( u ε ) ε> ∈ W ( M ) , such that (cid:13)(cid:13)(cid:13) u ε − u ε app | t=0 (cid:13)(cid:13)(cid:13) W ( M ) C ε, for some C > independent of ε . Then there exists ε ( T ) > , such that forany < ε ε ( T ) , the exact solution to the Cauchy problem (5.2) satisfies u ε ∈ L ∞ ([0 , T ]; W ( M )) . In addition, u ε app approximates u ε up to O ( ε ) : (cid:13)(cid:13) u ε − u ε app (cid:13)(cid:13) L ∞ ([0 ,T ] ×M ) (cid:13)(cid:13) u ε − u ε app (cid:13)(cid:13) L ∞ ([0 ,T ]; W ( M )) Cε, where C is independent of ε . Obviously the result for x ∈ T , announced in the introduction, can be seen as aspecial case of Theorem 6.5. Proof.
From Proposition 5.8, we may consider a solution u ε ∈ C ([0 , T ε ] , W ( M ))to (1.6). We define the difference w ε := u ε − u ε app . Then w ε ∈ C ([0 , τ ε ] , W ( M )),where τ ε = min( T ε , T ). We prove that for ε sufficiently small, w ε may be extendedup to time T , with w ε ∈ C ([0 , T ] , W ( M )). Take ε > C ε ≤ /
2, andfor ε ∈ ]0 , ε ], let t ε := sup n t ∈ [0 , T ] | sup t ′ ∈ [0 ,t ] k w ε ( t ′ ) k W ( M ) o . We already know that t ε > u ε . By possiblyreducing ε >
0, we shall show that t ε > T . The error term w ε solves: i∂ t w ε + ε w ε = λ (cid:0) | u ε app + w ε | σ ( u ε app + w ε ) − | u ε app | σ u ε app (cid:1) + λr ε − εr ε , where r ε , r ε are given in (6.2)–(6.3). Using Duhamel’s formula we can rewrite thisequation as w ε ( t ) = U ε ( t ) w ε − iλ Z t U ε ( t − τ ) (cid:0) | u ε app + w ε | σ ( u ε app + w ε ) − | u ε app | σ u ε app (cid:1) ( τ ) dτ − iλR ε ( t ) + iε Z t U ε ( t − τ ) r ε ( τ ) dτ, where R ε is defined in (6.5). Using the fact that U ε ( t ) is unitary on W ( M ), andthe estimates given in (6.4) and in Lemma 6.4, we obtain on [0 , t ε ]: k w ε ( t ) k W ( M ) C ε + | λ | Z t k (cid:0) | u ε app + w ε | σ ( u ε app + w ε ) − | u ε app | σ u ε app (cid:1) ( τ ) k W ( M ) dτ C ε + C Z t k w ε ( τ ) k W ( M ) dτ, by the Lipschitz property from Lemma 5.5 . Note that, in view of Lemma 5.2 , resp.Lemma 5.4, ( u ε app ) ε> is a bounded family in C ([0 , T ] , W ( M )), and restricting t to [0 , t ε ] ensures that w ε ( t ) stays bounded in W ( M ). The constants C , C dependon C and u ε app . Now, Gronwall lemma yields k w ε ( t ) k W ( M ) C ε (cid:18) e C T C (cid:19) , and we may reduce ε so that C ε (cid:0) e C T /C (cid:1) <
1. This shows that t ε > T ,for all ε ∈ ]0 , ε ]. Then, T ε > T follows, as well as the desired approximation of u ε by u ε app , since w ε = O ( ε ) in L ∞ ([0 , T ]; W ). (cid:3) Proof of the instability result
This section is devoted to the proof of Theorem 1.2. To this end we essentiallyrewrite the proof of M. Christ, J. Colliander, and T. Tao [8] in terms of weaklynonlinear geometric optics. It then becomes easy to see that the justification givenin the previous paragraph makes it possible to extend the one-dimensional analysisof [8] in order to infer Theorem 1.2.
Proof of Theorem 1.2.
We start with two Fourier modes, one of them being zero: i∂ t u + 12 ∆ u = λ | u | σ u ; u (0 , x ) = α + α e iKx , K ∈ N . The fact that we privilege oscillations with respect to the first space variable ispurely arbitrary. Define e u as the solution to the same equation, with data e u (0 , x ) = e α + e α e iKx . Let ε = 1 K ; u ε ( t, x ) = u (cid:18) t, x √ ε (cid:19) = u ( t, Kx ) . ( ε is chosen so that we remain on the torus.) We see that u ε solves (1.6) on T d ,with u ε (0 , x ) = α + α e ix /ε . From Theorem 6.5, we know that there exists
T > ε , such that k u ε − u ε app k L ∞ ([0 ,T ] × T d ) + k e u ε − e u ε app k L ∞ ([0 ,T ] × T d ) = O ( ε ) , where u ε app is the approximate solution defined by (6.1), and e u ε app is defined simi-larly. On the other hand, we have u ε app ( t, x ) = α e − iλtθ + α e − iλtθ e i ( x − t/ /ε , where, in view of (4.3), θ is given by θ = σ X n =0 (cid:18) σ + 1 n (cid:19) (cid:18) σn (cid:19) | α | σ − nl | α | n . We infer, uniformly in t ∈ [0 , T ], (cid:12)(cid:12)(cid:12)(cid:12)Z T d ( u ( t, x ) − e u ( t, x )) dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) α e − iλtθ − e α e − iλt e θ (cid:12)(cid:12)(cid:12) + O ( ε ) , with obvious notations.To prove the first point of Theorem 1.2, set α = e α = ρ , α = ρ K s = ρ K | s | , e α = r α + 1 δ . ULTIPHASE GEOMETRIC OPTICS FOR NLS 25
We infer, for 0 < δ (cid:12)(cid:12)(cid:12) θ − e θ (cid:12)(cid:12)(cid:12) & δ . We have k u (0) − e u (0) k H s < δ provided K > δ /s . Since e α = α , we also have (cid:12)(cid:12)(cid:12)(cid:12)Z T d ( u ( t, x ) − e u ( t, x )) dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) α sin (cid:18) λt (cid:16)e θ − θ (cid:17)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + O ( ε ) . We infer that we can find t ∈ [0 , δ ] so that the right hand side is bounded frombelow by ρ/
2, provided N is sufficiently large (hence ε sufficiently small).To prove the second point of Theorem 1.2, set α = ρ , e α = α + δ, α = e α = ρ K s = ρ K | s | . For δ small compared to ρ , we use the same estimate as above, (cid:12)(cid:12)(cid:12)(cid:12)Z T d ( u ( t, x ) − e u ( t, x )) dx (cid:12)(cid:12)(cid:12)(cid:12) & (cid:12)(cid:12)(cid:12)(cid:12) α sin (cid:18) λt (cid:16)e θ − θ (cid:17)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , for K sufficiently large. We now have (cid:12)(cid:12)(cid:12) θ − e θ (cid:12)(cid:12)(cid:12) & (cid:12)(cid:12) | α | σ − | e α | σ (cid:12)(cid:12) + | α | σ − (cid:12)(cid:12) | α | − | e α | (cid:12)(cid:12) & δ + (cid:16) ρK | s | (cid:17) σ − δ. Now we see that if we assume σ >
2, the left hand side can be estimated from belowby 1 /δ , provided N is sufficiently large, and we conclude like for the first point.To prove the last point in Theorem 1.2, we resume the argument of [25]. Fix α ∈ C \ { } , and let α ∈ C to be fixed later. As K → ∞ , we have: u (0 , · ) ⇀ α =: u (0 , · ) weakly in L ( T d ) ; k u (0) k L → | α | + | α | . For any t >
0, we have, as K → ∞ , u ( t, x ) ⇀ α e − iλtθ weakly in L ( T d ) , where θ = σ X n =0 (cid:18) σ + 1 n (cid:19) (cid:18) σn (cid:19) | α | σ − n | α | n . Note that for any α ∈ C \ { } and any angle θ ∈ [0 , π [, we can find α ∈ C sothat θ = θ + | α | σ . On the other hand, the solution to (1.8) with initial data α is given by u ( t, x ) = α e − iλt | α | σ . We infer w − lim N →∞ u ( t, x ) − u ( t, x ) = α e − iλt | α | σ (cid:0) e − iλtθ − (cid:1) . For all t = 0, one can then choose θ so that λtθ π Z . The discontinuity at α ofthe map α u ( t ), from L ( T d ) equipped with its weak topology into (cid:0) C ∞ ( T d ) (cid:1) ∗ ,follows. (cid:3) Appendix A. A more general set of initial phases
We can actually replace Assumption 6.2 with the following more general one:
Assumption A.1.
There exist b > , c > such that for all ( ℓ , . . . , ℓ σ +1 ) ∈ N , δ ( ℓ , . . . , ℓ σ +1 ) ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ +1 X p =1 ( − p +1 κ ℓ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − σ +1 X p =1 ( − p +1 (cid:12)(cid:12) κ ℓ p (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) satisfies: δ ( ℓ , . . . , ℓ σ +1 ) > c h κ ℓ i − b . . . (cid:10) κ ℓ σ +1 (cid:11) − b . In §
6, we have considered the case b = 0. However, allowing constants b > Proposition A.2.
For all p ∈ N ∗ , there exist C, b > and Z ⊂ R dp with zeroLebesgue measure such that, for all ( κ , . . . , κ p ) ∈ R dp \ Z , the set ( κ j ) j ∈ J con-structed from these initial wave vectors { κ j } j ∈ J satisfies Assumption A.1.Proof. We shall prove that the above result holds when Assumption A.1 is replacedby the stronger one, where N is replaced by J σ +1 .All the wave vectors we consider belong to the group generated by { κ , . . . , κ p } .Thus, to each ℓ k ∈ J corresponds ( α k, , . . . , α k,p ) ∈ Z p , such that: κ ℓ k = α k, κ + · · · + α k,p κ p . With this notation, for all ( ℓ , . . . , ℓ σ +1 ) ∈ J σ +1 , we have: δ ( ℓ , . . . , ℓ σ +1 ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ +1 X k =1 ( − k +1 p X j =1 α k,j κ j (cid:12)(cid:12)(cid:12) + σ +1 X m =1 ( − m +1 (cid:12)(cid:12)(cid:12) p X j =1 α m,j κ j (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p X i,j =1 σ +1 X k,ℓ =1 ( − k + ℓ α k,i α ℓ,j − σ +1 X m =1 ( − m α m,i α m,j κ i · κ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Now, a standard Diophantine result (see e.g. [1, 11]) ensures that, for all choiceof ( κ i · κ j ) i,j p but in some subset of R p with measure zero, we have, for some b ′ > C ′ > ∀ ( β i,j ) i,j p ∈ Z p \ { } , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p X i,j =1 β i,j κ i · κ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > C ′ p X i,j =1 | β i,j | − b ′ . Such an estimate is then valid for almost all ( κ , . . . , κ p ) in ( R d ) p . We apply it with β i,j = σ +1 X k,ℓ =1 ( − k + ℓ α k,i α ℓ,j − σ +1 X m =1 ( − m α m,i α m,j , so that p X i,j =1 | β i,j | σ +1 X k,ℓ =1 | α k, · || α ℓ, · | σ + 1) p Y k =1 h α k, · i . ULTIPHASE GEOMETRIC OPTICS FOR NLS 27
Now, choosing κ , . . . , κ p Q -linearly independent (which is true almost surely), weget that there exists a constant c > ∀ α ∈ Q p , | α | + · · · + | α p | c d X j =1 | ( α κ + · · · + α p κ p ) j | . Increasing c if necessary, so that c >
1, we get, when κ ℓ k = α k, κ + · · · + α k,p κ p : h α k, · i c h κ ℓ k i . Finally, using the constants b ′ and C ′ from above, the desiredestimate follows with b = 2 b ′ and C = (2(2 σ + 1) c ) − b ′ C ′ . (cid:3) Under Assumption A.1 (which is fairly general for plane waves, in view of theabove proposition), we can easily adapt the analysis of §
6. Essentially, we have to(possibly) strengthen the assumptions on the initial profile, in the case of M = R d ,where we generalize Assumption 6.1 to: Assumption A.3. On M = R d , the initial amplitudes satisfy: ∀| β | , ( h κ j i b ∂ βx α j ) j ∈ J ∈ E ( R d ) , ∀| β | , ( h κ j i b ∂ βx α j ) j ∈ J ∈ E ( R d ) . From Proposition 5.12, these data produce a solution ( a j ) j ∈ J ∈ C ([0 , T ] , E ( M ))to the profile system (5.3). We consequently define the approximate solution u ε app as before u ε app ( t, x ) = X j ∈ J a εj ( t, x ) e iφ j ( t,x ) /ε , where the sequence ( a j ) j ∈ J is now such that (cid:16) h κ j i b ∂ βx a j (cid:17) j ∈ J ∈ C ([0 , T ] , E ( M )) , | β | , (cid:16) h κ j i b ∂ βx a j (cid:17) j ∈ J ∈ C ([0 , T ] , E ( M )) , | β | . We can then reproduce the analysis of §
6: Lemma 6.4 is still valid under Assump-tion A.1 and A.3, by straightforward verification. Then one just has to notice thatthis is the only step where the absence of small divisors plays a role in the proofof Theorem 6.5. Therefore, Theorem 6.5 remains valid under Assumption A.1 andA.3.
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ULTIPHASE GEOMETRIC OPTICS FOR NLS 29
CNRS, UMR 5149, Montpellier, F-34095
E-mail address : [email protected] (E. Dumas)
Univ. Grenoble 1, Institut Fourier, 100, rue des Math´ematiques-BP 74,38402 Saint Martin d’H`eres cedex, France
E-mail address : [email protected] (C. Sparber)
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