# Numerical Simulation of Soliton Propagation Behavior for the Fractional-in-Space NLSE with Variable Coefficients on Unbounded Domain

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## Abstract

**:**

## 1. Introduction

## 2. Description of Method

**Property**

**1.**

**Property**

**2.**

**Property**

**3.**

## 3. Numerical Experiment

**Experiment**

**1.**

**Experiment**

**2.**

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Logarithmic of the absolute error on different long-time domains of Experiment 1 ($\alpha =2$).

**Figure 2.**Logarithmic of the absolute error on different long-time domains of Experiment 1 ($\alpha =2,$ $h=0.1,L=80,\text{}N=512$).

**Figure 3.**Simulation result of FSPB for Experiment 1 on different $\alpha $ and different long-time domains. (

**a**) $\alpha =1.6$. (

**b**) Modulus at different fractional derivatives and $t=200$.

**Figure 6.**Simulation result of FSPB for Experiment 1 ($\alpha =1.8,\text{}L=64,\text{}N=512,\text{}h=0.01,\text{}T=200$).

**Figure 7.**Simulation result of FSPB for Experiment 1 ($\mathbf{u}(x,0)=exp\left(i\right(x-2\left)\right)sec(2x+6),\text{}\alpha =2,$ $L=64,\text{}N=512,\text{}h=0.01,\text{}T=100$).

**Figure 9.**Simulation result of FSPB for Experiment 2 at $\mathbf{u}(x,0)=sec(6+2x)exp\left(i\right(-2+x\left)\right),$ $\alpha =1.6,\text{}L=64,\text{}N=512,\text{}h=0.01,\text{}T=100$.

**Figure 10.**Numerical simulation result of FSPB for Experiment 2 at $\mathbf{u}(x,0)=sec(6+2x)exp\left(i\right(-2+x\left)\right),\text{}\alpha =1.8,\text{}L=64,\text{}N=512,\text{}h=0.01,\text{}T=100$.

**Figure 11.**Simulation result of FSPB of Experiment 2 at $\mathbf{u}(x,0)=sec\left(2x\right)$ when $\alpha =2,\text{}L=64,$ $N=512,\text{}h=0.01,\text{}T=100$.

**Figure 12.**Simulation result for Experiment 2 at $\alpha =1.2,x=0$, $\mathbf{u}(x,0)=sec(6+2x)exp\left(i\right(-2+x\left)\right),\text{}L=64,\text{}N=512,\text{}h=0.01,\text{}T=100$.

**Table 1.**Simulation result of FSPB of Experiment 2 at $\alpha =0.2,1.2,2$, $\mathbf{u}(x,0)=\frac{1}{\sqrt{3}}sec$$\left(\frac{x}{3}\right)exp\left(\frac{i({x}^{2}-1)}{6}\right)$, $L=64,\text{}N=512,\text{}h=0.01,\text{}T=100$.

$\mathit{\alpha}$ | Imaginary Part | Real Part | Modulus |
---|---|---|---|

$\alpha =0.2$ | |||

$\alpha =1.2$ | |||

$\alpha =2$ |

**Table 2.**Simulation result of FSPB of Experiment 1 at three initial conditions when $\alpha =2,\text{}L=64,$ $N=512,\text{}h=0.01,\text{}T=100$.

Initial Condition | Imaginary Part | Real Part | Modulus |
---|---|---|---|

$sec(-0.5x)$ | |||

$sec(-2x)$ | |||

$sec(6+2x)exp\left(i\right(-2+x\left)\right)$ |

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**MDPI and ACS Style**

Tian, F.; Wang, Y.; Li, Z.
Numerical Simulation of Soliton Propagation Behavior for the Fractional-in-Space NLSE with Variable Coefficients on Unbounded Domain. *Fractal Fract.* **2024**, *8*, 163.
https://doi.org/10.3390/fractalfract8030163

**AMA Style**

Tian F, Wang Y, Li Z.
Numerical Simulation of Soliton Propagation Behavior for the Fractional-in-Space NLSE with Variable Coefficients on Unbounded Domain. *Fractal and Fractional*. 2024; 8(3):163.
https://doi.org/10.3390/fractalfract8030163

**Chicago/Turabian Style**

Tian, Fengzhou, Yulan Wang, and Zhiyuan Li.
2024. "Numerical Simulation of Soliton Propagation Behavior for the Fractional-in-Space NLSE with Variable Coefficients on Unbounded Domain" *Fractal and Fractional* 8, no. 3: 163.
https://doi.org/10.3390/fractalfract8030163