GEOMETRIC OPTICS AND INSTABILITY FOR NLS AND DAVEY STEWARTSON MODELS

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


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