Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations
type: Item
instance of: scholarly article
title: Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations
zbMATH Open document ID: 1119.35088
author: Jaime Angulo Pava
published in: Journal of Differential Equations
publication date: 2007-04-19T00:00:00Z
review text: The paper revisits the previously studied problem of the stability of periodic solutions to the nonlinear Schrödinger (NLS) equation in the form of Jacobi's elliptic functions (dn and cn, hence called ``cnoidal'' and ``dnoidal'' waves). It is proved that some families of these solutions are stable against perturbations that have the same periodicity as the unperturbed solutions, and (nonlinearly) unstable against perturbations with double period (a period-doubling instability). In fact, the instability of all periodic solutions to the NLS solution was known before. It is stated that the stability of other families of periodic solutions is unknown, and approaches developed in earlier well-known papers on this issue are criticized. The author also considers periodic cnoidal-type solutions to the modified Korteweg-de Vries equations, and states that stability remains unknown. In addition, there are considered some recently published particular exact solutions to the NLS equation with an external potential which are also expressed in terms of elliptic functions. It is stated that some of them are stable, and some are not.
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K45
zbMATH DE Number: 5144839
zbMATH Keywords: cnoidal wave
zbMATH Keywords: period doubling
zbMATH Keywords: periodic traveling waves
zbMATH Keywords: Jacobian elliptic functions
zbMATH Keywords: nonlinear stability
reviewed by: B. A. Malomed
MaRDI profile type: Publication
full work available at URL: j.jde.2007.01.003
OpenAlex ID: W2075685495
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DOI: 10.1016/J.JDE.2007.01.003
description: scientific article; zbMATH DE number 5144839