Wave dynamics on networks: method and application to the sine-Gordon equation

TL;DR

This paper develops a symplectic finite difference method for solving nonlinear wave equations on networks, specifically applying it to the sine-Gordon equation to analyze wave dynamics at junctions.

Contribution

It introduces a novel numerical scheme for nonlinear wave equations on complex networks, with a detailed treatment of interface conditions at vertices.

Findings

Effective simulation of wave propagation on networks with loops.

Successful modeling of kinks and breathers in the sine-Gordon equation.

Demonstrated scheme's stability and accuracy in numerical tests.

Abstract

We consider a scalar Hamiltonian nonlinear wave equation formulated on networks; this is a non standard problem because these domains are not locally homeomorphic to any subset of the Euclidean space. More precisely, we assume each edge to be a 1D uniform line with end points identified with graph vertices. The interface conditions at these vertices are introduced and justified using conservation laws and an homothetic argument. We present a detailed methodology based on a symplectic finite difference scheme together with a special treatment at the junctions to solve the problem and apply it to the sine-Gordon equation. Numerical results on a simple graph containing four loops show the performance of the scheme for kinks and breathers initial conditions.