Solutions of Fixed Period in the Nonlinear Wave Equation on Networks

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 $g:{\mathbb{R}}^n\to {\mathbb{R}}^n$ 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.