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GEOMETRIC OPTICS AND INSTABILITY FOR NLS AND DAVEY STEWARTSON MODELS

33 pages
GEOMETRIC OPTICS AND INSTABILITY FOR NLS AND DAVEY-STEWARTSON MODELS REMI CARLES, ERIC DUMAS, AND CHRISTOF SPARBER Abstract. We study the interaction of (slowly modulated) high frequency waves for multi-dimensional nonlinear Schrodinger equations with gauge invari- ant power-law nonlinearities and non-local perturbations. The model includes the Davey–Stewartson system in its elliptic-elliptic and hyperbolic-elliptic vari- ant. Our analysis reveals a new localization phenomenon for non-local pertur- bations in the high frequency regime and allows us to infer strong instability results on the Cauchy problem in negative order Sobolev spaces, where we prove norm inflation with infinite loss of regularity by a constructive approach. Contents 1. Introduction 2 1.1. Motivation 2 1.2. Weakly nonlinear geometric optics 4 1.3. Instability and norm inflation 6 2. Interaction of high frequency waves in NLS type models 8 2.1. Geometric optics for elliptic NLS 8 2.2. Geometric optics for the DS system 11 2.3. Possible generalizations 14 3. Construction of the exact and approximate solutions 16 3.1. Analytical framework 16 3.2. Existence results 16 4. Justification of multiphase geometric optics 18 4.1. Localizing the non-local oscillations 19 4.2. Filtering the non-characteristic oscillations 20 4.3. Proof of Theorem 4.1 23 5. More weakly nonlinear geometric optics 24 5.1. Approximate solution 24 5.2. Negligible or not? 25 6.

  • weakly nonlinear

  • well posedness

  • heuristically expect local

  • expect global

  • unique solution

  • local perturbation

  • waves within

  • ?j ?


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GEOMETRIC OPTICS AND INSTABILITY FOR NLS DAVEY-STEWARTSON MODELS
´ REMI CARLES, ERIC DUMAS, AND CHRISTOF SPARBER
AND
Abstract.interaction of (slowly modulated) high frequencyWe study the wavesformulti-dimensionalnonlinearSchr¨odingerequationswithgaugeinvari-ant power-law nonlinearities and non-local perturbations. The model includes the Davey–Stewartson system in its elliptic-elliptic and hyperbolic-elliptic vari-ant. Our analysis reveals a new localization phenomenon for non-local pertur-bations in the high frequency regime and allows us to infer strong instability results on the Cauchy problem in negative order Sobolev spaces, where we prove norm inflation with infinite loss of regularity by a constructive approach.
Contents
1. Introduction 1.1. Motivation 1.2. Weakly nonlinear geometric optics 1.3. Instability and norm inflation 2. Interaction of high frequency waves in NLS type models 2.1. Geometric optics for elliptic NLS 2.2. Geometric optics for the DS system 2.3. Possible generalizations 3. Construction of the exact and approximate solutions 3.1. Analytical framework 3.2. Existence results 4. Justification of multiphase geometric optics 4.1. Localizing the non-local oscillations 4.2. Filtering the non-characteristic oscillations 4.3. Proof of Theorem 4.1 5. More weakly nonlinear geometric optics 5.1. Approximate solution 5.2. Negligible or not? 6. Norm inflation 6.1. Scaling 6.2. High frequency analysis 6.3. Proof of Theorem 1.10 6.4. Proof of Theorem 1.12 Appendix A. Proof of Proposition 1.8 Appendix B. On negative order Sobolev spaces References
2 2 4 6 8 8 11 14 16 16 16 18 19 20 23 24 24 25 28 28 28 29 30 30 31 32
This work was supported by the French ANR project R.A.S. (ANR-08-JCJC-0124-01) and by the Royal Society Research fellowship of C. Sparber. 1
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R. CARLES, E. DUMAS, AND C. SPARBER
1.onnItroducti
1.1..MoitavitnoTheDavey-Stewartson system(DS) provides a canonical de-scription of the dynamics of weakly nonlinear two-dimensional waves interacting with a mean-fieldχ(t x1 x2)R; see [18] for more details. the following we In shall consider DS)i∂tψ2+1ηx2x211++2xx222χψ==λxx11χ|ψ|+2µ|ψ|2ψ  ( where (x1 x2)xR2,tR, andλ µRare some given parameters. In addition, the choiceη=±1 distinguishes between to the so-calledctiipll-eicptlile and theyphboerc-liillecitp the DSvariants of the DS system (see [18]). Clearly, system withη= +1 andλ= 0 simplifies to the cubiconlincSrhenragnreo¨id equation(NLS), which we consider more generally in thed-dimensional case
i∂tψΔ21+ψ=µ|ψ|2ψ  xRd. The cubic NLS equation is a canonical model for weakly nonlinear wave propagation in dispersive media and has numerous applications in e.g. nonlinear optics, quantum superfluids, or the description of water waves, cf. [18]. We shall allow for more general, gauge invariant, nonlinearities and consider (NLS)i∂tψ+Δ21ψ=µ|ψ|2νψ  xRd ? whereνN such equations, one usually distinguishes between. Forfocusing µ <0 andgincosudfeµ >0 nonlinearities. The sign ofµhas a huge impact on the issue of global well-posedness, since it is well known (cf. [18] for a broad review) that forµ <0finite-time blow-upof solutions may occur ford>2, that is: tliTmkrψ(t)kL2(Rd)= T<+.
Thus, in general we cannot expect global well-posedness to hold in, say,H1(Rd). On the other hand, one might ask about the possibility that even local (in time) well-posedness fails. To be more precise, we recall the following definition:
Definition 1.1.Letσ sR. The Cauchy problem for(1.6)is well posed from Hs(Rd)toHσ(Rd)if, for all bounded subsetBHs(Rd), there existT >0and a Banach spaceXT,C([0 T];Hσ(Rn))such that: (1)For allϕBH,(1.6)has a unique solutionψXTwithψ|t=0=ϕ. (2)The mappingϕ(BHk∙kHs)7→ψC([0 T];Hσ(Rn))is continuous. The negation of the above definition is called alack of well-posednessorin-stability order to gain a rough idea why instability occurs, we consider the. In Cauchy problem of (NLS) with initial dataψ0Hs(Rd the assumption). Under νN?nonlinearity is smooth, and thus, local well-posedness (from, the Hs(Rd) toHs(Rd)) holds for sufficiently larges(s > d/2 does the job). On the other hand, one should note that (NLS) is invariant underGalilei transformations, ψ(t x)7→eivxi|v|2t/2ψ(t xvt) vRd
GEOMETRIC OPTICS AND INSTABILITY FOR NLS AND DAVEY-STEWARTSON 3 which leave theL2(Rd) norm invariant. In addition, solutions to (NLS) are invari-ant under thescaling symmetry ψ(t x)7→Λ1ψΛt2ΛxΛ>0.
Denoting
1 (1.1)s2=:dcν this scaling is easily seen to leave the homogeneous Sobolev spaceH˙sc(Rd) invariant and thus we heuristically expect local well-posedness to hold only inHs(Rd) with s>max{sc0} reason for this being that for. Thes <max{sc0}and sufficiently large Λ>0 we can use the scaling symmetry of (NLS) to relate the norm of large solutions at timet >0 to the norm of small solutions at some timet< t other. In words, the difference between two solutions will immediately become very big in Hs(Rdeven if they are close to each other initially.), For the cubic NLS equation we havesc= 0 ifd= 2, and thus instability should occur forψ0Hs(R2) withs < we expect the same behavior to be0. Moreover, true also for the DS system, since the latter can be written in form of an NLS with non-local perturbation, i.e. 2 2 (1.2)i∂tψ12+η∂2x1+x22ψ=λE|ψ|ψ+µ|ψ|2 xψ R. Here the operatorEacting as a Fourier multiplier on|ψ|2is defined via (1.[ξ21b 3)E(f)(ξ) =ξ12+ξ22f(ξ)where (ξ1 ξ2) =ξR2andfbdenotes the Fourier transform off, defined as (1.4) (Ff)(ξ)fb(ξ) = (2π)1d/2ZRdf(x)eixξdx. With this normalization, we haveF1g=Fgˇ, with ˇg=g(−∙). Note that the non-local term in (1.2) scales like the nonlinearity in the cubic NLS equation, since the kernel bξ12L(R2)(1.5)K(ξ) =ξ12+ξ22 ishomogeneous of degree zero. We therefore expect instability of the DS system in Sobolev spaces of negative order. It will be one of the main tasks of this work to rigorously prove this type of instability, which can be seen as a negative result, complementing the well-posedness theorems of [12] (see also [11]). To this end, we shall rely on the framework ofweakly nonlinear geometric optics(WNLGO), developed in [5] for NLS. We shall henceforth study, as a first step, the interaction ofhighly oscillatory waveswithin (1.2) and describe the possible (nonlinear)reso-nances our opinion this is interesting in itself since it generalizes Inbetween them. the results of [5] and reveals a new localization property for non-local operators in the high frequency regime. Moreover, we shall see that possible resonances heavily depend on the choice ofη=±1.