Multi-solitary waves for the nonlinear Klein-Gordon equation

TL;DR

This paper proves the existence of multi-solitary wave solutions for the nonlinear Klein-Gordon equation in multiple dimensions, constructed as sums of stable boosted standing waves, using advanced analytical techniques.

Contribution

It introduces a rigorous method to construct multi-solitary waves for the nonlinear Klein-Gordon equation based on stability and approximation techniques.

Findings

Existence of multi-solitary wave solutions under stability conditions.

Use of backward-in-time analysis and convergence to exact solutions.

Application of variational and energy methods in the proof.

Abstract

We consider the nonlinear Klein-Gordon equation in $\R^d$. We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy estimates.