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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2015
- DOI:
- 10.48550/arXiv.1506.02405
- arXiv:
- arXiv:1506.02405
- Bibcode:
- 2015arXiv150602405D
- Keywords:
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- Mathematical Physics;
- Mathematics - Analysis of PDEs;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- Physics - Computational Physics
- E-Print:
- 31 pages, 9 figures, 2 tables, 41 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/