Trees, trunks, and branches -- bifurcation structure of time-periodic solutions to $u_{tt}-u_{xx}\pm u^{3}=0$

TL;DR

This paper analyzes the bifurcation structure of time-periodic solutions to a cubic wave equation, revealing complex interactions and suggesting solutions exist at any frequency, regardless of size.

Contribution

It introduces a systematic approach to study the bifurcation structure of solutions, complementing previous existence proofs and highlighting the role of mode interactions.

Findings

Solutions exist for any frequency.

Solutions can be arbitrarily large.

Complex mode interactions are key to solution structure.

Abstract

We propose a systematic approach to analysing the complex structure of time-periodic solutions to the cubic wave equation on an interval with Dirichlet boundary conditions first reported in arXiv:2407.16507. The analysis we present is based on a detailed study of sparse mode interactions suggested by the previous numerical work. Our results complement prior rigorous existence proofs and suggest that solutions exist for any frequency, however, they may be arbitrarily large.