# Nonlinear Management of Topological Solitons in a Spin-Orbit-Coupled System

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## Abstract

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## 1. Introduction

## 2. The Model and Analytical Results

## 3. Results: Stability Regions for Solitons under the Action of the Management

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The ground-state soliton of the MM (mixed-mode) type, produced as a numerical solution of Equation (1) with ${\gamma}_{1}=0$ in Equation (6), for ${\gamma}_{0}=1.1$. (

**b**) A ground-state soliton of the SD (semi-dipole) type, for ${\gamma}_{0}=0.9$. The solid and dashed lines show the profiles of $|{\varphi}_{+}\left(x\right)|$ and $|{\varphi}_{-}\left(x\right)|$ respectively.

**Figure 2.**Shaded are stability regions in the parameter plane of $({\gamma}_{0},\omega )$ for (

**a**) MM- and (

**b**) SD-type solitons at a fixed management amplitude, ${\gamma}_{1}=0.05$ in Equation (6). Solitons are unstable in blank areas.

**Figure 3.**(

**a**) The evolution of the norm ratio $R\left(t\right)$ (see Equation (14)) for the MM, initially perturbed as per Equation (13), with $\delta =0.01$, at ${\gamma}_{0}=1.1$ and ${\gamma}_{1}=0$. (

**b**) The evolution of $R\left(t\right)$ (solid line) and $\langle x\rangle $ (dashed line) are shown, along with the underlying modulation format $\gamma \left(t\right)$ (Equation (6), the dotted line), for the MM initiated by input (13) with $\delta =0.001$, at ${\gamma}_{0}=1.1$, ${\gamma}_{1}=0.05$, and $\omega =0.68$. (

**c**) The same as in (

**b**), but for $\omega =0.335$.

**Figure 4.**The evolution of the center-of-mass position, $\langle x\rangle $, for the SD solitons, shown along with the underlying time-periodic modulation $\gamma \left(t\right)$, with ${\gamma}_{0}=0.95$ and ${\gamma}_{1}=0.12$ in Equation (6), for three values of the modulation frequency: (

**a**) $\omega =0.31$, (

**b**) $\omega =0.28$, and (

**c**) $\omega =0.25$. Panel (

**d**) illustrates the unstable evolution of the soliton corresponding to (

**c**).

**Figure 5.**Shaded are stability regions in the parameter plane of the management parameters, $({\gamma}_{1},\omega )$ (see Equation (6)) for MM-type solitons at fixed ${\gamma}_{0}=1.05$ (

**a**) and SD states (

**b**) at ${\gamma}_{0}=0.95$.

**Figure 6.**Stability boundaries for the SD and MM dynamical states (solid and dashed lines, respectively) at ${\gamma}_{1}=0.05$ (

**a**), ${\gamma}_{1}=0.1$ (

**b**), and ${\gamma}_{1}=0.15$ (

**c**). The bistability holds between the boundaries, in the vertically-shaded areas (see the text).

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Sakaguchi, H.; Malomed, B.
Nonlinear Management of Topological Solitons in a Spin-Orbit-Coupled System. *Symmetry* **2019**, *11*, 388.
https://doi.org/10.3390/sym11030388

**AMA Style**

Sakaguchi H, Malomed B.
Nonlinear Management of Topological Solitons in a Spin-Orbit-Coupled System. *Symmetry*. 2019; 11(3):388.
https://doi.org/10.3390/sym11030388

**Chicago/Turabian Style**

Sakaguchi, Hidetsugu, and Boris Malomed.
2019. "Nonlinear Management of Topological Solitons in a Spin-Orbit-Coupled System" *Symmetry* 11, no. 3: 388.
https://doi.org/10.3390/sym11030388