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.

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

References in corpus (4)

• Numerical simulation of a solitonic gas in KdV and KdV-BBM equations
• Nonlinear waves in networks: a simple approach using the sine-Gordon equation
• Shape waves in 2D Josephson junctions: exact solutions and time dilation
• On supraconvergence phenomenon for second order centered finite differences on non-uniform grids

Cited by in corpus (6)

• Noise-induced, ac-stabilized sine-Gordon breathers: Emergence and statistics
• Coupling Conditions for Water Waves at Forks
• Unstable kink-soliton profiles for the sine-Gordon equation on a $\mathcal{Y}$-junction graph with $\delta$-interaction
• Universal quantum Hawking evaporation of integrable two-dimensional solitons
• An Asymptotic-Preserving Scheme for Isentropic Flow in Pipe Networks
• Spectral solutions of PDEs on networks