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arXiv · 1210.1322

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

David Chiron
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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.

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David Chiron. 2012-10-04. Stability and instability for subsonic travelling waves of the Nonlinear Schrödinger Equation in dimension one. https://doi.org/10.2140/apde.2013.6.1327

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