Dynamics of Klein-Gordon on a compact surface near an homoclinic orbit

TL;DR

This paper investigates the behavior of solutions to the Klein-Gordon equation on a compact surface near a homoclinic orbit, demonstrating the existence of many non-small solutions that remain close to this orbit over time.

Contribution

It introduces a new analysis of the Klein-Gordon dynamics near a homoclinic orbit on a compact surface, extending previous methods to construct non-small solutions.

Findings

Existence of many solutions close to the homoclinic orbit

Solutions are not necessarily small in amplitude

The approach adapts Groves-Schneider strategy to this setting

Abstract

We consider the Klein-Gordon equation on a Riemannian surface which is globally well-posed in the energy space. This equation has an homoclinic orbit to the origin, and in this paper we study the dynamics close to it. Using a strategy from Groves-Schneider, we show that there are many solutions which stay close to this homocline during all times. We point out that the solutions we construct are not small.