Stability and instability for subsonic travelling waves of the Nonlinear Schr\"odinger Equation in dimension one

TL;DR

This paper provides a comprehensive analysis of the stability and instability of subsonic travelling waves in the one-dimensional Nonlinear Schrödinger Equation, introducing methods for stability proofs and instability detection.

Contribution

It develops systematic methods for stability and instability analysis, including constructing Lyapunov functionals and applying spectral theories, covering critical cases like kink solutions and energy-momentum cusps.

Findings

Complete stability/instability classification in energy space.

Construction of Lyapunov functionals for stability proofs.

Identification of unstable eigenvalues via Evans function.

Abstract

We study the stability/instability of the subsonic travelling waves of the Nonlinear Schr\"odinger 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.