Solutions of Fixed Period in the Nonlinear Wave Equation on Networks

TL;DR

This paper proves the existence of multiple symmetric, non-constant periodic solutions for a nonlinear wave equation on networks, using topological methods and symmetry considerations.

Contribution

It introduces a novel application of equivariant topological invariants to establish multiple periodic solutions in network wave equations.

Findings

Multiple non-constant p-periodic solutions exist

Solutions are characterized by their symmetries

Method applies to networks with permutation invariance

Abstract

The wave equation on network is defined by $\partial_{tt}u=\Delta_{G}u+g(u)$, where $u\in\mathbb{R}^{n}$ and the graph Laplacian $\Delta_{G}$ is an operator on functions on $n$ vertices. We suppose that $g:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is an odd continuous function that satisfies $g(0)=g^{\prime }(0)=0$ and the Nagumo condition. Assuming that the graph is invariant by a subgroup of permutations $\Gamma$, using a $\Gamma$-equivariant topological invariant we prove the existence of multiple non-constant $p$-periodic solutions characterized by their symmetries.