Wavepacket preservation under nonlinear evolution
| dc.creator | Babin, A. | |
| dc.creator | Figotin, A. | |
| dc.date | 2006-07-27 | |
| dc.date | 2007-04-10 | |
| dc.date.accessioned | 2026-07-07T07:55:51Z | |
| dc.date.available | 2026-07-07T07:55:51Z | |
| dc.description | 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. | |
| dc.description | References added, typos corrected | |
| dc.identifier | https://arxiv.org/abs/math/0607723 | |
| dc.identifier | http://arxiv.org/abs/math/0607723 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/127105 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Mathematical Physics | |
| dc.subject | 35Q55; 35Q60; 35L70; 35L75; 35B34; 35B40 | |
| dc.title | Wavepacket preservation under nonlinear evolution | |
| dc.type | text |