## Abstract

We consider the interactions of traveling localized wave solutions with a vertex in a star graph domain that describes multiple Josephson junctions with a common/branch point (i.e., tricrystal junctions). The system is modeled by the sine-Gordon equation. The vertex is represented by boundary conditions that are determined by the continuity of the magnetic field and vanishing total fluxes. When one considers small-amplitude breather solutions, the system can be reduced into the nonlinear Schrödinger equation posed on a star graph. Using the equation, we show that a high-velocity incoming soliton is split into a transmitted component and a reflected one. The transmission is shown to be in good agreement with the transmission rate of plane waves in the linear Schrödinger equation on the same graph (i.e., a quantum graph). In the context of the sine-Gordon equation, small-amplitude breathers show similar qualitative behaviors, while large-amplitude ones produce complex dynamics.

Original language | British English |
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Article number | 271 |

Journal | Symmetry |

Volume | 11 |

Issue number | 2 |

DOIs | |

State | Published - 1 Feb 2019 |

## Keywords

- Breather
- Quantum graph
- Schrödinger equation
- Sine-Gordon equation
- Soliton
- Star graph