A review on asymptotic stability of solitary waves in nonlinear dispersive problems in dimension one

Germain, Pierre

Journées équations aux dérivées partielles (2024), Exposé no. 6, 22 p.

We review asymptotic stability of solitary waves for nonlinear dispersive equations set on the line. Our focus is threefold: first, the nonlinear Schrödinger equation; second, the notion of full asymptotic stability (which states that perturbations of a solitary wave decompose globally into a solitary wave and a decaying solution); and third, spectral methods. Besides this focus, we summarize the state of the art in a broader context, including nonlinear Klein–Gordon equations, the notion of local asymptotic stability, and virial methods.

Cet article de survol s’intéresse à la stabilité asymptotique des ondes solitaires d’équations dispersives non-linéaires. Nous nous attacherons plus particulièrement à l’équation de Schrödinger non-linéaire, à la notion de stabilité asymptotique complète (qui demande que la solution se décompose asymptotiquement en une onde solitaire et une radiation décroissante) et aux méthodes spectrales. Nous tenterons aussi de présenter l’état de l’art dans un contexte plus général, incluant l’équation de Klein–Gordon non-linéaire, la notion de stabilité asymptotique locale et les méthodes de viriel.

Keywords: nonlinear dispersive equations, solitary waves, asymptotic stability
Mots-clés : équations dispersives non linéaires, ondes solitaires, stabilité asymptotique

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