Asymptotic stability and classification of multi-solitons for Klein-Gordon equations

TL;DR

This paper establishes the asymptotic stability and classifies multi-soliton solutions for Klein-Gordon equations, showing convergence to superpositions of solitons and constructing a finite-dimensional manifold of such solutions.

Contribution

It introduces new stability results, classification of multi-solitons, and a finite-codimension manifold structure for solutions of Klein-Gordon equations.

Findings

Multi-solitons are asymptotically stable under certain conditions.

Solutions close to multi-solitons scatter to superpositions of Lorentz-transformed solitons.

A finite-dimensional manifold of multi-solitons is constructed.

Abstract

Focusing on multi-solitons for the Klein-Gordon equations, in first part of this paper, we establish their conditional asymptotic stability. In the second part of this paper, we classify pure multi-solitons which are solutions converging to multi-solitons in the energy space as $t\rightarrow\infty$. Using Strichartz estimates developed in our earlier work \cite{CJ2} and the modulation techniques, we show that if a solution stays close to the multi-soliton family, then it scatters to the multi-soliton family in the sense that the solution will converge in large time to a superposition of Lorentz-transformed solitons (with slightly modified velocities), and a radiation term which is at main order a free wave. Moreover, we construct a finite-codimension centre-stable manifold around the well-separated multi-soliton family. Finally, given different Lorentz parameters and arbitrary centers, we show that all the corresponding pure multi-solitons form a finite-dimension manifold.