Localized solutions of nonlinear network wave equations

TL;DR

This paper investigates localized solutions in nonlinear network wave equations, approximating their dynamics with Duffing equations and analyzing conditions for localization based on network parameters and eigenfrequencies.

Contribution

It introduces a novel approximation method for localized solutions using Duffing equations and analyzes localization conditions in nonlinear network wave systems.

Findings

Localization amplitude depends on maximal normal eigenfrequency.

Approximation by Duffing equation accurately models large amplitude localized initial conditions.

Numerical analysis identifies parameter regimes for localization.

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 Schrodinger 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.