Continuum limit of the discrete nonlinear Klein-Gordon equation

TL;DR

This paper investigates how solutions of the discrete nonlinear Klein-Gordon equation behave as the lattice spacing approaches zero, using advanced mathematical tools to establish convergence and discuss implications for nonlinear wave analysis.

Contribution

It introduces a novel approach combining Shannon interpolation and discrete Sobolev norm controls to analyze the continuum limit of the discrete nonlinear Klein-Gordon equation.

Findings

Proves convergence of discrete solutions to the continuum equation

Develops bilinear estimates for Shannon interpolation

Provides insights into uniform dispersive estimates for nonlinear waves

Abstract

We study the convergence of solutions of the discrete nonlinear Klein-Gordon equation on an infinite lattice in the continuum limit, using recent tools developed in the context of nonlinear discrete dispersive equations. Our approach relies in particular on the use of bilinear estimates of the Shannon interpolation alongside controls on the growth of discrete Sobolev norms of the solution. We conclude by giving perspectives on uniform dispersive estimates for nonlinear waves on lattices.