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Stability and instability for subsonic travelling waves of the Nonlinear Schrö̈dinger 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.
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Submitted on : Friday, October 18, 2013 - 11:28:50 AM
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David Chiron. Stability and instability for subsonic travelling waves of the Nonlinear Schrö̈dinger Equation in dimension one. Analysis and Partial Differential Equations, 2013, 6 (6), pp.1327-1420. ⟨hal-00874585⟩

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