New class of time-periodic solutions to the 1D cubic wave equation

TL;DR

This paper rigorously constructs a new class of time-periodic solutions for the 1D defocusing cubic wave equation, confirming previous numerical suggestions through verified operator bounds.

Contribution

It introduces a rigorous construction method for these solutions, utilizing rational arithmetic to verify key operator bounds, advancing understanding of wave equation solutions.

Findings

Confirmed existence of new time-periodic solutions

Used rational arithmetic for verification

Established operator bounds rigorously

Abstract

In recent papers (arXiv:2407.16507, arXiv:2408.05158) we presented results suggesting the existence of a new class of time-periodic solutions to the defocusing cubic wave equation on a one-dimensional interval with Dirichlet boundary conditions. Here we confirm these findings by rigorously constructing solutions from this class. The proof uses rational arithmetic computations to verify essential operator bounds.