Stability and instability for subsonic travelling waves of the Nonlinear Schrödinger Equation in dimension one
نویسنده
چکیده
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 GrillakisShatah-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. Key-words: travelling wave, Nonlinear Schrödinger Equation, Gross-Pitaevskii Equation, stability, Evans function, Liapounov functional. MSC (2010): 35B35, 35C07, 35J20, 35Q40, 35Q55.
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