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

摘要

We study the stability/instability of the subsonic travelling waves of theNonlinear Schr\"odinger Equation in dimension one. Our aim is to proposeseveral methods for showing instability (use of the Grillakis-Shatah-Strausstheory, proof of existence of an unstable eigenvalue via an Evans function) orstability. For the later, we show how to construct in a systematic way aLiapounov functional for which the travelling wave is a local minimizer. Theseapproaches allow to give a complete stability/instability analysis in theenergy space including the critical case of the kink solution. We also treatthe case of a cusp in the energy-momentum diagram.