  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 negative order Sobolev spaces,
where we prove norm inflation with infinite loss of regularity by a
constructive approach.
