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

Denys Dutykh, Jean Guy Caputo

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

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.

Original languageBritish English
Pages (from-to)54-71
Number of pages18
JournalApplied Numerical Mathematics
Volume131
DOIs
StatePublished - Sep 2018

Keywords

  • Graph theory
  • HAMILTONIAN partial differential equations
  • Partial differential equations on networks
  • Sine-GORDON equation

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