Long time stability of some small amplitude solutions in nonlinear Schrödinger equations
type: Item
instance of: scholarly article
title: Long time stability of some small amplitude solutions in nonlinear Schrödinger equations
zbMATH Open document ID: 0895.35095
author: Dario Bambusi
published in: Communications in Mathematical Physics
publication date: 1998-09-27T00:00:00Z
review text: The author considers small perturbations of the Zakharov-Shabat nonlinear Schrödinger equation on \([0,2M]\) with vanishing or periodic boundary conditions in \([0,M]\). It is given a bound on the long time diffusion of the actions for initial data in open subsets of the phase space in the spirit of Nekhoroshev's theorem. The result restricts to solutions which (1) are of small amplitude, and (2) correspond to initial data close to the majority of finite dimensional invariant tori of the linearized system, namely those fulfilling a suitable (generic) nonresonance condition, with the above limitations. The difficulties associated to the actual integration of the ZS equation are avoided, and, the result concerns precisely the dynamics in a neighbourhood of nonresonant tori.
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 1121194
zbMATH Keywords: small perturbations of the Zakharov-Shabat nonlinear Schrödinger equation
zbMATH Keywords: Nekhoroshev's theorem
zbMATH Keywords: linearized system
reviewed by: L. Vázquez
MaRDI profile type: Publication
full work available at URL: s002200050196
OpenAlex ID: W1970348793
DOI: 10.1007/S002200050196
description: scientific article; zbMATH DE number 1121194