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.

Keywords

Bibliographic record

BibTeX Citation

@article{Dutykh2018wavedynamics,
  author = {Dutykh, D. and Caputo, J.-G.},
  title = {Wave dynamics on networks: method and application to the sine—Gordon equation},
  journal = {Appl. Numer. Math.},
  year = {2018},
  volume = {131},
  pages = {54--71},
  doi = {10.1016/j.apnum.2018.03.010},
  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.},
  keywords = {graph theory, Partial differential equations on networks, sine-Gordon equation, Hamiltonian partial differential equations}
}