Wavepacket preservation under nonlinear evolution

Abstract

We study nonlinear systems of hyperbolic (in a wider sense) PDE's in entire d-dimensional space describing wave propagation with the initial data in the form of a finite sum of wavepackets referred to as multi-wavepackets. The problem involves two small parameters beta and rho where: (i) (1/beta) is a factor describing spatial extension of the wavepackets; (ii) (1/rho) is a factor describing the relative magnitude of the linear part of the evolution equation compared to its nonlinearity. For a wide range of the small parameters and on time intervals long enough for strong nonlinear effects we prove that multi-wavepackets are preserved under the nonlinear evolution. In particular, the corresponding wave vectors and the band numbers of involved wavepackets are "conserved quantities". We also prove that the evolution of a multi-wavepacket is described with high accuracy by a properly constructed system of envelope equations with a universal nonlinearity which in simpler cases turn into well-known Nonlinear Schrodinger or coupled modes equations. The universal nonlinearity is obtained by a certain time averaging applied to the original nonlinearity. This can be viewed as an extension of the well known averaging method developed for finite-dimensional nonlinear oscillatory systems to the case of a general translation invariant PDE systems with the linear part having continuous spectrum.