Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation.(English)Zbl 1179.37101

This tutorial (or mini-monograph) deals with orbital stability of the standing waves viewed as solutions of the form $$e^{t\lambda A}\varphi$$, $$\lambda\in \mathbb{R}$$, $$\varphi\in X$$, of some infinite dimensional Hamiltonian system, X being an infinite dimensional space. The theory is applied to the case of the nonlinear Schrödinger equation whose standing waves are time-harmonic. The problems discussed are: the Hamiltonian system and its invariance, sufficient stability conditions for orbital stability of a standing wave, constrained minimization and stability, the nonlinear Schrödinger equation, the nonlinear Schrödinger equation with a power law nonlinearity.

MSC:

 37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems 37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory 35Q55 NLS equations (nonlinear Schrödinger equations)
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