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

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.Comment: 31 pages, 9 figures, 2 tables, 41 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

Multi-symplectic spectral methods for the sine-Gordon equation

Recently it has been shown that spectral discretizations provide another class of multi-symplectic integrators for Hamiltonian wave equations with periodic boundary conditions. In this note we develop multi-symplectic spectral discretizations for the sine-Gordon equation. We discuss the preservation of its phase space geometry, as measured by the associated nonlinear spectrum, by the multi-symplectic spectral methods