Perturbations of the defocusing nonlinear Schrödinger equation
type: Item
instance of: scholarly article
title: Perturbations of the defocusing nonlinear Schrödinger equation
zbMATH Open document ID: 1048.37067
published in: Milan Journal of Mathematics
publication date: 2004-09-07T00:00:00Z
review text: The authors study the KAM theory for the defocusing nonlinear Schrödinger (NLS) equation \[ i\partial_t\varphi=-\partial^2_x\varphi+2| \varphi| ^2\varphi, \quad \varphi(x+1,t)=\varphi(x,t), \quad x,t\in {\mathbb{R}}. \] The basic theorem says that many of the NLS-invariant tori (not necessarily close to the zero solution) persist and remain linearly stable under small perturbations of the Hamiltonian satisfying three natural conditions. Main technical tool is a detailed treatment of the frequencies \(\omega_k\). In particular, new closed formulas for \(\omega_k\) are provided which are of independent interest. The purpose of the present paper is to document that some results and methods from \textit{T. Kappeler} and \textit{J. Pöschel} [KdV \& KAM, Berlin: Springer (2003; Zbl 1032.37001)] can applied to the NLS equation, too. The main results of the paper were previously announced by the authors in [J. Nonlinear Math. Phys. 8, Suppl., 133--138 (2001; Zbl 0977.35133].
reviewed by: Iliya D. Iliev
Mathematics Subject Classification ID: 37K55
Mathematics Subject Classification ID: 35Q55
zbMATH DE Number: 2099231
zbMATH Keywords: KAM theory
zbMATH Keywords: persistence of invariant tori
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: small perturbations
author: Benoît Grébert
author: T. Kappeler
MaRDI profile type: Publication
full work available at URL: s00032-002-0018-2
OpenAlex ID: W1994206189
DOI: 10.1007/S00032-002-0018-2
description: scientific article; zbMATH DE number 2099231