Wave dynamics on networks: method and application to the sine-Gordon equation
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.
Domains
Analysis of PDEs [math.AP] Mathematical Physics [math-ph] Computational Physics [physics.comp-ph] Exactly Solvable and Integrable Systems [nlin.SI] Fluid mechanics [physics.class-ph] Numerical Analysis [math.NA] Numerical Analysis [cs.NA] Modeling and Simulation Pattern Formation and Solitons [nlin.PS]
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