A note on small periodic solutions of discrete nonlinear Klein-Gordon equations

TL;DR

This paper demonstrates the existence of small time periodic solutions in discrete nonlinear Klein-Gordon equations with potential, contrasting the continuous case where such solutions are generally absent.

Contribution

The paper establishes the existence of small periodic solutions for discrete nonlinear Klein-Gordon equations, a result not applicable to the continuous case.

Findings

Existence of small time periodic solutions in discrete equations

Contrast with continuous Klein-Gordon equations

Potential-dependent solutions identified

Abstract

In this note, we consider discrete nonlinear Klein-Gordon equations with potential. By the pioneering work of Sigal, it is known that for the "continuous" nonlinear Klein-Gordon equation, no small time periodic solution exists generically. However, for the discrete nonlinear Klein-Gordon equations, we show that there exist small time periodic solutions.