Journal of the European Mathematical Society

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Volume 14, Issue 6, 2012, pp. 1885–1921
DOI: 10.4171/JEMS/350

Geometric optics and instability for NLS and Davey–Stewartson models

Rémi Carles[1], Eric Dumas[2] and Christof Sparber[3]

(1) CNRS et Université Montpellier 2, Mathématiques, CC 051, Place Eugène Bataillon, 34095, MONTPELLIER CEDEX 5, FRANCE
(2) Institut Fourier, Université Grenoble I, B.P. 74, F-38402, SAINT MARTIN D'HERES CEDEX, FRANCE
(3) Department of Mathematics, University of Illinois at Chicago, 851 S. Morgan Street, IL 60607-7045, CHICAGO, UNITED STATES

We study the interaction of (slowly modulated) high frequency waves for multi-dimensional nonlinear Schr¨odinger equations with gauge invariant power-law nonlinearities and nonlocal perturbations. The model includes the Davey–Stewartson system in its elliptic-elliptic and hyperbolic-elliptic variants. Our analysis reveals a new localization phenomenon for nonlocal perturbations 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.

Keywords: Nonlinear Schrödinger equation, Davey–Stewartson system, geometric optics, instability

Carles Rémi, Dumas Eric, Sparber Christof: Geometric optics and instability for NLS and Davey–Stewartson models. J. Eur. Math. Soc. 14 (2012), 1885-1921. doi: 10.4171/JEMS/350