Localized solutions of nonlinear network wave equations

Abstract

We study localized solutions for the nonlinear graph wave equation on finite arbitrary networks. Assuming a large amplitude localized initial condition on one node of the graph, we approximate its evolution by the Duffing equation. The rest of the network satisfies a linear system forced by the excited node. This approximation is validated by reducing the nonlinear graph wave equation to the discrete nonlinear Schrödinger equation and by Fourier analysis. Finally, we examine numerically the condition for localization in the parameter plane, coupling versus amplitude and show that the localization amplitude depends on the maximal normal eigenfrequency.