Multi-solitons for nonlinear Klein-Gordon equations

TL;DR

This paper constructs multi-soliton solutions for the nonlinear Klein-Gordon equation in high-dimensional space, extending previous methods to the wave case and controlling unstable modes without modulation theory.

Contribution

It introduces a novel approach to build multi-solitons for NLKG, overcoming instability issues and generalizing linear theory to boosted solitons.

Findings

Existence of N-soliton solutions in energy space for large times.

Extension of linear stability theory to boosted solitons.

New solutions applicable to recent Nakanishi-Schlag analysis.

Abstract

In this paper we consider the existence of multi-soliton structures for the nonlinear Klein-Gordon equation (NLKG) in R^{1+d}. We prove that, independently of the unstable character of (NLKG) solitons, it is possible to construct a N-soliton family of solutions to (NLKG), of dimension 2N, globally well-defined in the energy space H^1 \times L^2 for all large positive times. The method of proof involves the generalization of previous works on supercritical NLS and gKdV equations by Martel, Merle and the first author to the wave case, where we replace the unstable mode associated to the linear NLKG operator by two generalized directions that are controlled without appealing to modulation theory. As a byproduct, we generalize the linear theory described in Grillakis-Shatah-Strauss and Duyckaerts-Merle to the case of boosted solitons, and provide new solutions to be studied using the recent Nakanishi- Schlag theory.