The stationary sine-Gordon equation on metric graphs: Exact analytical solutions for simple topologies

TL;DR

This paper derives exact analytical solutions for the stationary sine-Gordon equation on simple metric graph topologies, with potential applications in Josephson junctions and branched superconducting devices.

Contribution

It introduces a method to solve the stationary sine-Gordon equation on metric graphs with simple topologies, extending to more complex structures.

Findings

Exact solutions for star graph topologies

Method extension to tree and other simple graphs

Applications to Josephson junctions and tricrystal boundaries

Abstract

We consider the stationary sine-Gordon equation on metric graphs with simple topologies. The vertex boundary conditions are provided by flux conservation and matching of derivatives at the star graph vertex. Exact analytical solutions are obtained. It is shown that the method can be extended for tree and other simple graph topologies. Applications of the obtained results to branched planar Josephson junctions and Josephson junctions with tricrystal boundaries are discussed.