On existence and uniqueness of asymptotic $N$-soliton-like solutions of the nonlinear klein-gordon equation

TL;DR

This paper establishes the existence and uniqueness of solutions to the nonlinear Klein-Gordon equation that asymptotically resemble sums of solitons, with detailed convergence properties and classifications for single and multiple soliton cases.

Contribution

It provides the first comprehensive analysis of multi-soliton solutions for the NLKG, including exponential convergence and uniqueness results.

Findings

Constructed an N-parameter family of solutions converging to soliton sums.

Proved uniqueness of solutions for N=1 within a specific class.

Established exponential convergence rates for multi-soliton solutions.

Abstract

We are interested in solutions of the nonlinear Klein-Gordon equation (NLKG) in $\mathbb{R}^{1+d}$, $d\ge1$, which behave as a soliton or a sum of solitons in large time. In the spirit of other articles focusing on the supercritical generalized Korteweg-de Vries equations and on the nonlinear Schr{\"o}dinger equations, we obtain an $N$-parameter family of solutions of (NLKG) which converges exponentially fast to a sum of given (unstable) solitons. For $N = 1$, this family completely describes the set of solutions converging to the soliton considered; for $N\ge 2$, we prove uniqueness in a class with explicit algebraic rate of convergence.