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arXiv · 1506.02405

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

Denys Dutykh·Jean-Guy Caputo
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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.

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Denys Dutykh, Jean-Guy Caputo. 2015-06-08. Wave dynamics on networks: method and application to the sine-Gordon equation. https://doi.org/10.1016/j.apnum.2018.03.010

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