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

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.