Vol. 6, No. 6, 2013

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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

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.

traveling wave, nonlinear Schrödinger equation, Gross–Pitaevskii equation, stability, Evans function, Liapounov functional
Mathematical Subject Classification 2010
Primary: 35B35, 35J20, 35Q40, 35Q55, 35C07
Received: 25 June 2012
Revised: 4 September 2012
Accepted: 28 February 2013
Published: 18 November 2013
David Chiron
Laboratoire J.A. Dieudonné
Université de Nice-Sophia Antipolis
Parc Valrose
06108 Nice Cedex 02