Small Amplitude Periodic Solutions of Klein-Gordon Equations

TL;DR

This paper constructs small amplitude periodic solutions for a class of nonlinear Klein-Gordon equations with different temporal and spatial scales, expanding understanding of their oscillatory behaviors.

Contribution

It introduces a method to find small amplitude periodic solutions with distinct temporal and spatial periods for nonlinear Klein-Gordon equations.

Findings

Existence of small amplitude periodic solutions.

Solutions have different temporal and spatial scales.

Method applicable to a class of nonlinear Klein-Gordon equations.

Abstract

We consider a class of nonlinear Klein-Gordon equations $u_{tt}=u_{xx}-u+f(u)$ and obtain a family of small amplitude periodic solutions, where the temporal and spatial period have different scales.