Global Attraction to Solitary Waves in Models Based on the Klein-Gordon Equation

TL;DR

This paper reviews recent findings on the long-term behavior of Klein-Gordon-based models, showing that under certain conditions, their global attractors are primarily composed of solitary waves, including multifrequency solutions.

Contribution

It demonstrates that in U(1)-invariant dispersive Hamiltonian systems based on the Klein-Gordon equation, the weak global attractor is characterized by solitary waves, expanding understanding of their asymptotic dynamics.

Findings

Global attractors are mainly solitary waves under generic assumptions.

Examples of systems with multifrequency solitary waves are provided.

Theoretical framework for the asymptotic behavior of Klein-Gordon models is developed.

Abstract

We review recent results on global attractors of U(1)-invariant dispersive Hamiltonian systems. We study several models based on the Klein-Gordon equation and sketch the proof that in these models, under certain generic assumptions, the weak global attractor is represented by the set of all solitary waves. In general, the attractors may also contain multifrequency solitary waves; we give examples of systems which contain such solutions.