Scattering theory of the nonlinear wave equations on lattices

TL;DR

This paper develops a scattering theory for nonlinear wave equations on lattices, combining decay estimates and dispersive analysis to prove existence of wave operators and asymptotic completeness.

Contribution

It introduces a comprehensive scattering framework for discrete nonlinear wave equations, integrating decay estimates and Strichartz estimates for the first time.

Findings

Established uniform decay estimates for discrete wave equations.

Proved existence of wave operators for DNLW.

Demonstrated asymptotic completeness using multiple methods.

Abstract

In this paper, I will summarize the uniform decay estimates of the discrete wave equations (DW) established by the oscillatory integral theory in [Sch98, BCH23, BCH24], and combine the abstract framework of the scattering theory of the dispersive equations established in [Str81] to finally establish the scattering theory of the discrete nonlinear wave equations (DNLW), including the existence of the wave operators and the asymptotic completeness. In addition, I will establish the scattering theory again by the Strichartz estimate.