Decomposition of global solutions for a class of nonlinear wave equations

TL;DR

This paper proves the existence of free channel wave operators and the decomposition of solutions into free and localized parts for a class of nonlinear wave equations, extending scattering theory results.

Contribution

It extends scattering results to nonlinear wave equations by establishing wave operator existence and solution decomposition under boundedness and localization assumptions.

Findings

Existence of free channel wave operators for the nonlinear wave equations.

Decomposition of solutions into free and localized components when the nonlinearity is localized.

Extension of scattering theory results from Schrödinger to nonlinear wave equations.

Abstract

In the present paper we consider global solutions of a class of non-linear wave equations of the form \begin{equation*} \Box u= N(x,t,u)u, \end{equation*} where the nonlinearity~$ N(x,t,u)u$ is assumed to satisfy appropriate boundedness assumptions. Under these appropriate assumptions we prove that the free channel wave operator exists. Moreover, if the interaction term~$N(x,t,u)u$ is localised, then we prove that the global solution of the full nonlinear equation can be decomposed into a `free' part and a `localised' part. The present work can be seen as an extension of the scattering results of~\cite{SW20221} for the Schr\"odinger equation.