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

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.