Stability and instability for subsonic traveling waves of the nonlinear Schrödinger equation in dimension one

Chiron, David

doi:10.2140/apde.2013.6.1327

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Stability and instability for subsonic traveling waves of the nonlinear Schrödinger equation in dimension one

David Chiron

Vol. 6 (2013), No. 6, 1327–1420

DOI: 10.2140/apde.2013.6.1327

Abstract

We study the stability/instability of the subsonic traveling waves of the nonlinear Schrödinger equation in dimension one. Our aim is to propose several methods for showing instability (use of the Grillakis–Shatah–Strauss theory, proof of existence of an unstable eigenvalue via an Evans function) or stability. For the latter, we show how to construct in a systematic way a Liapounov functional for which the traveling wave is a local minimizer. These approaches allow us to give a complete stability/instability analysis in the energy space including the critical case of the kink solution. We also treat the case of a cusp in the energy-momentum diagram.

Keywords

traveling wave, nonlinear Schrödinger equation, Gross–Pitaevskii equation, stability, Evans function, Liapounov functional

Mathematical Subject Classification 2010

Primary: 35B35, 35J20, 35Q40, 35Q55, 35C07

Milestones

Received: 25 June 2012

Revised: 4 September 2012

Accepted: 28 February 2013

Published: 18 November 2013

Authors

David Chiron
Laboratoire J.A. Dieudonné Université de Nice-Sophia Antipolis Parc Valrose 06108 Nice Cedex 02 France

Vol. 6, No. 6, 2013