Time-periodic solutions to the cubic wave equation: an elementary constructive approach

TL;DR

This paper introduces an elementary constructive method to prove the existence of infinite time-periodic solutions to the 1D cubic wave equation, providing explicit details about their frequencies and structures.

Contribution

It offers a new, straightforward proof using perturbative expansion and Banach contraction, differing from prior approaches by giving explicit solution characteristics.

Findings

Existence of infinite time-periodic solutions established.

Explicit frequencies and structures of solutions obtained.

Method simplifies previous complex proofs.

Abstract

We present an elementary proof of existence of infinite family of time-periodic solutions to the one-dimensional nonlinear cubic wave equation with Dirichlet boundary conditions. It relies on the first order perturbative expansion and uses the Banach contraction principle to show existence of nearby solutions. In contrast to the previous results, this approach provides us explicit information about the frequencies and structures of the obtained solutions.