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We study the interaction of (slowly modulated) high frequency waves for multi-dimensional nonlinear Schrodinger equations with gauge invariant power-law nonlinearities and non-local perturbations. The model includes the Davey--Stewartson system in its elliptic-elliptic and hyperbolic-elliptic variant. Our analysis reveals a new localization phenomenon for non-local perturbations in the high frequency regime and allows us to infer strong instability results on the Cauchy problem in negativedoi:10.4171/jems/350 fatcat:d4nxtqw6ard2lmpjlnscflyhou