Global attraction to solitary waves for Klein-Gordon equation with mean field interaction

TL;DR

This paper proves that solutions to a nonlinear Klein-Gordon equation with mean field interaction in one or more dimensions tend to a set of nonlinear eigenfunctions over time, due to energy transfer and dispersive effects.

Contribution

It establishes the global attraction to nonlinear eigenfunctions for the Klein-Gordon equation with mean field interaction, under generic conditions, extending understanding of long-time dynamics.

Findings

Solutions converge to nonlinear eigenfunctions over time

Energy transfer causes dispersive radiation leading to attraction

Results apply to equations in one or more spatial dimensions

Abstract

We consider a U(1)-invariant nonlinear Klein-Gordon equation in dimension one or larger, self-interacting via the mean field mechanism. We analyze the long-time asymptotics of finite energy solutions and prove that, under certain generic assumptions, each solution converges (as time goes to infinity) to the two-dimensional set of all ``nonlinear eigenfunctions'' of the form $\phi(x)e\sp{-i\omega t}$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.