Periodic Solutions for the 1d Cubic Wave Equation with Dirichlet Boundary Conditions

TL;DR

This paper investigates time-periodic solutions of the 1D cubic wave equation with Dirichlet boundary conditions, combining rigorous perturbative analysis with numerical exploration of complex bifurcation structures.

Contribution

It provides a rigorous fourth-order expansion derivation and uncovers intricate bifurcation patterns of solutions, linking numerical findings to theoretical conjectures.

Findings

Existence of time-periodic solutions for arbitrary frequencies.

Discovery of fractal-like bifurcation structures in solution space.

Numerical evidence supporting Cantor set families of solutions.

Abstract

We study time-periodic solutions for the cubic wave equation on an interval with Dirichlet boundary conditions. We begin by following the perturbative construction of Vernov and Khrustalev and provide a rigorous derivation of the fourth-order expansion in small amplitude, which we use to verify the Galerkin scheme. In the main part, we focus on exploring large solutions numerically. We find an intricate bifurcation structure of time-periodic solutions forming a fractal-like pattern and explore it for the first time. Our results suggest that time-periodic solutions exist for arbitrary frequencies, with appearance of fine bifurcation structure likely related to the Cantor set families of solutions described in previous rigorous works.