Global Attraction to Solitary Waves

Abstract

We study properties of solitary wave solutions of the form $\phi(x)e^{-i\omega t}$, with ω real and $\phi(x)$) localized in space. In the first section, we sketch two fundamental results on stability of solitary waves: Derrick’s theorem on instability of time-independent solutions and the Vakhitov–Kolokolov stability criterion for spectral stability of solitary waves.

The main subject of the article is the structure of the (weak) global attractor of finite energy solutions to nonlinear Hamiltonian systems. The solitary resolution conjecture states that such an attractor is formed by the set of all solitary waves.We give the proof of this result for the simplest model: the Klein–Gordon field in one spatial dimension, coupled to a nonlinear oscillator. The main building block of the proof is the Titchmarsh convolution theorem; for completeness, we provide its proof.

Mathematics Subject Classification (2010). Primary 35B41; Secondary 37K40.