# Soliton and breather splitting on star graphs from tricrystal Josephson junctions

## TL;DR

This paper investigates how localized wave solutions, including solitons and breathers, interact with a vertex in a star graph domain modeling tricrystal Josephson junctions, revealing splitting and transmission behaviors consistent with quantum graph theory.

## Contribution

It introduces a reduced model using the nonlinear Schrödinger equation to analyze soliton splitting and transmission in Josephson junction star graphs, connecting nonlinear wave dynamics with quantum graph concepts.

## Key findings

- High-velocity solitons split into transmitted and reflected parts.
- Transmission rates align with linear Schrödinger equation predictions.
- Large-amplitude breathers exhibit complex, nonlinear behaviors.

## 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 Schr\"odinger 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 Schr\"odinger 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.