Stability and instability for subsonic travelling waves of the Nonlinear Schrö̈dinger Equation in dimension one

David Chiron (1)

1 JAD - Laboratoire Jean Alexandre Dieudonné

We study the stability/instability of the subsonic travelling 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 later, we show how to construct in a systematic way a Liapounov functional for which the travelling wave is a local minimizer. These approaches allow 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.

Domains: Analysis of PDEs [math.AP]

David Chiron. Stability and instability for subsonic travelling waves of the Nonlinear Schrö̈dinger Equation in dimension one. Analysis & PDE, 2013, 6 (6), pp.1327-1420. ⟨hal-00874585⟩