Sine-Gordon solitons in networks: Scattering and transmission at vertices

TL;DR

This paper analyzes sine-Gordon solitons on network graphs, deriving boundary conditions from conservation laws, and demonstrates reflectionless transmission of solitons at vertices, with applications to various physical systems.

Contribution

It introduces a method to derive boundary conditions for sine-Gordon equations on graphs and finds conditions for reflectionless soliton transmission, extending integrability to network structures.

Findings

Analytical solutions for star and tree graphs show reflectionless soliton transmission.

A sum rule for coefficients ensures complete integrability on the graph.

Numerical solutions quantify vertex scattering in general cases.

Abstract

We consider the sine-Gordon equation on metric graphs with simple topologies and derive vertex boundary conditions from the fundamental conservation laws, such as energy and current conservation. Traveling wave solutions for star and tree graphs are obtained analytically in the form of kink, antikink and breather solitons for a special case. It is shown that these solutions provide reflectionless soliton transmission at the graph vertex. We find the sum rule for bond-dependent coefficients making the sine-Gordon equation embedded on the graph completely integrable. For the general case the problem is solved numerically and the vertex scattering is quantified. Applications of the obtained results to Josephson junction networks, DNA double helix and elastic fibre networks are discussed.