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.
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Rémi Carles, Eric Dumas, Christof Sparber, Geometric optics and instability for NLS and Davey–Stewartson models. J. Eur. Math. Soc. 14 (2012), no. 6, pp. 1885–1921DOI 10.4171/JEMS/350