Analysis & PDE

  • Anal. PDE
  • Volume 6, Number 6 (2013), 1327-1420.

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

David Chiron

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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.

Article information

Anal. PDE, Volume 6, Number 6 (2013), 1327-1420.

Received: 25 June 2012
Revised: 4 September 2012
Accepted: 28 February 2013
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 35B35: Stability 35J20: Variational methods for second-order elliptic equations 35Q40: PDEs in connection with quantum mechanics 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35C07: Traveling wave solutions

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


Chiron, David. Stability and instability for subsonic traveling waves of the nonlinear Schrödinger equation in dimension one. Anal. PDE 6 (2013), no. 6, 1327--1420. doi:10.2140/apde.2013.6.1327.

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