Validity of the nonlinear Schr\"odinger approximation for quasilinear dispersive systems with more than one derivative

Abstract

For nonlinear dispersive systems, the nonlinear Schr\"odinger (NLS) equation can usually be derived as a formal approximation equation describing slow spatial and temporal modulations of the envelope of a spatially and temporally oscillating underlying carrier wave. Here, we justify the NLS approximation for a whole class of quasilinear dispersive systems, which also includes toy models for the waterwave problem. This is the first time that this is done for systems, where a quasilinear quadratic term is allowed to effectively lose more than one derivative. With effective loss we here mean the loss still present after making a diagonalization of the linear part of the system such that all linear operators in this diagonalization have the same regularity properties.