# Soliton and Breather Splitting on Star Graphs from Tricrystal Josephson Junctions

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Scattering of NLS Solitons

#### 3.1. Scattering in the Linear Problems

#### 3.2. NLS Soliton Scattering

## 4. Scattering of sG Breathers

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NLS | nonlinear Schrödinger |

sG | sine-Gordon |

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**Figure 1.**A schematic diagram of the system, showing a breather traveling from $x\ll 0$ towards $x=0$. The convention of the spatial direction used herein is indicated.

**Figure 2.**(

**a**,

**b**) Dynamics of a soliton moving towards the branch point $X=0$ with the initial velocity $v=0.3$ (

**a**) and $v=3$ (

**b**). Shown are the top view of ${|U(X,T)|}^{2}$. (

**c**) Numerically obtained transmission rate ${\tilde{T}}_{s}$ as a function of the incoming soliton velocity v. The horizontal dashed line is the theoretical approximation ${\tilde{t}}^{2}=\frac{4}{9}$ from (11).

**Figure 3.**Similar to Figure 2, but for (small-amplitude) sG breathers with velocity $v=0.1$ (

**a**) and $v=0.3$ (

**b**). In panels (

**a**,

**b**), shown are the top view of $H(x,t)$. Panel (

**c**) is the transmission rate ${\tilde{T}}_{b}$ as a function of the incoming breather velocity v. The horizontal dashed line is the theoretical approximation ${\tilde{t}}^{2}=\frac{4}{9}$ from (11). Here, $\theta ={cos}^{-1}0.99$.

**Figure 4.**Similar to Figure 3a,b, but for a large-amplitude sG breather with velocity $v=0.1$ (

**a**) and $v=0.5$ (

**b**). Here, $\theta ={cos}^{-1}0.1$.

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**MDPI and ACS Style**

Susanto, H.; Karjanto, N.; Zulkarnain; Nusantara, T.; Widjanarko, T.
Soliton and Breather Splitting on Star Graphs from Tricrystal Josephson Junctions. *Symmetry* **2019**, *11*, 271.
https://doi.org/10.3390/sym11020271

**AMA Style**

Susanto H, Karjanto N, Zulkarnain, Nusantara T, Widjanarko T.
Soliton and Breather Splitting on Star Graphs from Tricrystal Josephson Junctions. *Symmetry*. 2019; 11(2):271.
https://doi.org/10.3390/sym11020271

**Chicago/Turabian Style**

Susanto, Hadi, Natanael Karjanto, Zulkarnain, Toto Nusantara, and Taufiq Widjanarko.
2019. "Soliton and Breather Splitting on Star Graphs from Tricrystal Josephson Junctions" *Symmetry* 11, no. 2: 271.
https://doi.org/10.3390/sym11020271