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

Abstract

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.

83 pages

References in corpus (4)

• Low-dimensional Bose liquids: beyond the Gross-Pitaevskii approximation
• Orbital stability of the black soliton to the Gross-Pitaevskii equation
• Dark Solitons in Discrete Lattices: Saturable versus Cubic Nonlinearities
• Dark solitons in external potentials

Cited by in corpus (3)

• Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, II
• Small energy traveling waves for the Euler-Korteweg system
• Periodic traveling waves for nonlinear Schrödinger equations with non-zero conditions at infinity in ${\mathbb{R}}^2$