ar X iv : m at h / 06 07 72 3 v 1 [ m at h . A P ] 2 7 Ju l 2 00 6 Wavepacket preservation under nonlinear evolution

We study nonlinear systems of hyperbolic (in a wider sense) PDE's in R d 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 β and ̺ where: (i) 1 β is a factor describing spatial extension of the wavepackets; (ii) 1 ̺ 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.