# Soliton Solution of Schrödinger Equation Using Cubic B-Spline Galerkin Method

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equation and Cubic B-Spline Galerkin Method

## 3. Stability Analysis

## 4. Numerical Results and Test Problems

#### 4.1. Single Soliton Solution to the NLS Equation

#### 4.2. The Interaction of Two Solitons for the NLS Equation

#### 4.3. Birth of Standing Soliton with the Maxwellian Initial Condition

**,**otherwise the soliton will decay away [22].

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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$\Delta \mathit{t}=0.0025$ | $\Delta \mathit{t}=0.005$ | |||||||
---|---|---|---|---|---|---|---|---|

$\mathit{t}$ | ${\mathit{L}}_{\mathit{\infty}}$ | ${\mathit{L}}_{2}$ | ${\mathit{I}}_{1}$ | ${\mathit{I}}_{2}$ | ${\mathit{L}}_{\mathit{\infty}}$ | ${\mathit{L}}_{2}$ | ${\mathit{I}}_{1}$ | ${\mathit{I}}_{2}$ |

0.5 | 0.0000896 | 0.0000914 | 2.0 | 7.33333332 | 0.00018 | 0.00018 | 2.0 | 7.33333327 |

1 | 0.0000896 | 0.0000914 | 2.0 | 7.33333332 | 0.00018 | 0.00018 | 2.0 | 7.33333327 |

1.5 | 0.0000896 | 0.0000914 | 2.0 | 7.33333332 | 0.00018 | 0.00018 | 2.0 | 7.33333327 |

2 | 0.0000896 | 0.0000914 | 2.0 | 7.33333331 | 0.00018 | 0.00018 | 2.0 | 7.33333327 |

2.5 | 0.0000896 | 0.0000914 | 2.0 | 7.33333330 | 0.00018 | 0.00018 | 2.0 | 7.33333325 |

3 | 0.0000896 | 0.0000914 | 2.0 | 7.33333223 | 0.00018 | 0.00018 | 2.0 | 7.33333219 |

3.5 | 0.0000896 | 0.0000914 | 1.99999 | 7.33327430 | 0.00018 | 0.00018 | 1.99999 | 7.33327425 |

**Table 2.**Comparisons of the present results with those of Taha et al. [20]., amplitude = 1 at time = 1.

$\mathit{h}$ | $\Delta \mathit{t}$ | ${\mathit{L}}_{\mathit{\infty}}$ | ${\mathit{L}}_{2}$ | ${\mathit{I}}_{1}$ | ${\mathit{I}}_{2}$ | ${\mathit{L}}_{\mathit{\infty}}$ [20] |
---|---|---|---|---|---|---|

0.05 | 0.0025 | 0.0001086 | 0.0001106 | 2.0 | 7.33333330 | 0.008 |

0.05 | 0.000625 | 0.0000272 | 0.0000277 | 2.0 | 7.33333333 | 0.006 |

0.05 | 0.001 | 0.0000435 | 0.0000443 | 2.0 | 7.33333333 | 0.006 |

0.08 | 0.002 | 0.0000628 | 0.0000639 | 2.0 | 7.33333330 | 0.005 |

0.3125 | 0.00026 | 0.0003174 | 0.0004973 | 2.00001 | 7.33332035 | 0.005 |

0.3125 | 0.020 | 0.0024417 | 0.0038252 | 2.00010 | 7.33320599 | 0.005 |

0.05 | 0.0005 | 0.0000217 | 0.0000221 | 2.0 | 7.33333333 | 0.008 |

0.3125 | 0.0026 | 0.0003174 | 0.0004973 | 2.00001 | 7.33332035 | 0.006 |

**Table 3.**Rate of convergence in spatial and temporal directions at $t=1\text{}\mathrm{with}\text{}h=\Delta t.$

$\mathit{h}$ | ${\mathit{L}}_{\mathit{\infty}}$ | Order | ${\mathit{L}}_{2}$ | Order |
---|---|---|---|---|

0.050000 | 0.0020492 | $-$ | 0.0019146 | - |

0.025000 | 0.0005730 | 1.83 | 0.0005302 | 1.85 |

0.012500 | 0.0001507 | 1.93 | 0.000139 | 1.93 |

0.001625 | 0.0000386 | 1.97 | 0.0000356 | 1.96 |

0.003125 | 0.00000965 | 2.00 | 0.0000089 | 2.00 |

**Table 4.**Error norms and conservation laws for single soliton with $h=0.05,\Delta t=0.0025$ $h=0.05,q=2,S=4,\beta =2$.

$\mathit{t}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{\mathit{\infty}}$ |
---|---|---|

0.5 | 0.0005580 | 0.0007595 |

1 | 0.0005580 | 0.0007595 |

1.5 | 0.0005580 | 0.0007595 |

2 | 0.0005580 | 0.0007595 |

2.5 | 0.0005580 | 0.0007595 |

3 | 0.0005580 | 0.0007595 |

3.5 | 0.0005580 | 0.0007595 |

4 | 0.0005580 | 0.0007595 |

**Table 5.**Comparisons of single soliton with those of Taha et al. [20], amplitude = 2 at time = 1.

$\mathit{h}$ | $\Delta \mathit{t}$ | ${\mathit{L}}_{\mathit{\infty}}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{\mathit{\infty}}$ [20] |
---|---|---|---|---|

0.02 | 0.0025 | 0.0003909 | 0.0002859 | 0.0011 |

0.066 | 0.002 | 0.0009658 | 0.0007168 | 0.006 |

0.02 | 0.0001 | 0.0000156 | 0.0000114 | 0.009 |

0.03 | 0.00022 | 0.0000143 | 0.0000105 | 0.008 |

0.1 | 0.0025 | 0.0016852 | 0.0012732 | 0.0004 |

0.1563 | 0.0011 | 0.0010143 | 0.0007852 | 0.008 |

$\mathit{t}$ | $\mathit{h}$ | $\Delta \mathit{t}$ | ${\mathit{L}}_{\mathit{\infty}}$ | ${\mathit{L}}_{2}$ |
---|---|---|---|---|

1 | 0.0667 | 0.0025 | 0.0001601 | 0.0002367 |

1.5 | 0.133 | 0.01 | 0.0012135 | 0.0018299 |

2 | 0.133 | 0.01 | 0.0011711 | 0.0017397 |

3.5 | 0.133 | 0.01 | 0.0012135 | 0.0018299 |

4.5 | 0.133 | 0.01 | 0.0012144 | 0.0018317 |

0.5 | 0.625 | 0.005 | 0.0018529 | 0.0030593 |

1 | 0.625 | 0.005 | 0.0017864 | 0.0030560 |

1 | 0.05 | 0.001 | 0.0000486 | 0.0000716 |

1.5 | 0.05 | 0.001 | 0.0000485 | 0.0000715 |

3 | 0.5 | 0.0036 | 0.0011024 | 0.0017556 |

3 | 0.5 | 0.0025 | 0.0007655 | 0.0012192 |

5 | 0.1 | 0.0025 | 0.0002342 | 0.0003495 |

**Table 7.**Comparisons of two solitons with Taha et al. [20], amplitude = 1.

Galerkin Cubic B-spline (Present Method) | Taha et al. [20] | |||
---|---|---|---|---|

t | $\mathit{h}$ | $\Delta \mathit{t}$ | ${\mathit{L}}_{\mathit{\infty}}$ | ${\mathit{L}}_{\mathit{\infty}}$ |

1.6 | 0.05 | 0.001 | 0.0000485 | 0.00173 |

1.8 | 0.07 | 0.07 | 0.0047374 | 0.00158 |

1 | 0.05 | 0.0025 | 0.0001214 | 0.00096 |

1 | 0.05 | 0.001 | 0.0000486 | 0.00141 |

1 | 0.625 | 0.0071 | 0.0025367 | 0.00122 |

1 | 0.130 | 0.0036 | 0.0004304 | 0.00141 |

**Table 8.**Comparison of conserved quantities of the birth of a standing soliton with Mokhtari et al. [11], $h=0.08,\Delta t=0.004\text{}\mathrm{and}\text{}q=2$.

Galerkin Cubic B-spline (Present Method) | Mokhtari et al. [11] | |||||
---|---|---|---|---|---|---|

$\mathit{t}$ | ${\mathit{I}}_{1}$ | ${\mathit{I}}_{2}$ | $\mathit{E}{\mathit{I}}_{1}$ | $\mathit{E}{\mathit{I}}_{2}$ | $\mathit{E}{\mathit{I}}_{1}$ | $\mathit{E}{\mathit{I}}_{2}$ |

$1$ | 3.97100052134110 | $-4.925146$ | 8.67067 × 10^{−9} | 4.7 × 10^{−4} | 3.5 × 10^{−11} | 4.8 × 10^{−10} |

$2$ | 3.97100050351461 | $-4.925347$ | 2.08442 × 10^{−8} | 2.7 × 10^{−4} | 3.8 × 10^{−11} | 6.9 × 10^{−10} |

$3$ | 3.97100050291892 | $-4.925138$ | 2.02485 × 10^{−8} | 4.8 × 10^{−4} | 1.1 × 10^{−10} | −6.4 × 10^{−9} |

$4$ | 3.9710005812876 | $-4.925325$ | 6.86172 × 10^{−8} | 2.9 × 10^{−4} | −2.0 × 10^{−8} | −4.5 × 10^{−6} |

$5$ | 3.97100055661924 | $-4.925367$ | 4.39488 × 10^{−8} | 2.5 × 10^{−4} | −9.2 × 10^{−7} | −4.7 × 10^{−5} |

$6$ | 3.97100053521529 | $-4.925432$ | 2.25449 × 10^{−8} | 1.9 × 10^{−4} | −6.0 × 10^{−6} | −9.8 × 10^{−5} |

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**MDPI and ACS Style**

Iqbal, A.; Abd Hamid, N.N.; Md. Ismail, A.I.
Soliton Solution of Schrödinger Equation Using Cubic B-Spline Galerkin Method. *Fluids* **2019**, *4*, 108.
https://doi.org/10.3390/fluids4020108

**AMA Style**

Iqbal A, Abd Hamid NN, Md. Ismail AI.
Soliton Solution of Schrödinger Equation Using Cubic B-Spline Galerkin Method. *Fluids*. 2019; 4(2):108.
https://doi.org/10.3390/fluids4020108

**Chicago/Turabian Style**

Iqbal, Azhar, Nur Nadiah Abd Hamid, and Ahmad Izani Md. Ismail.
2019. "Soliton Solution of Schrödinger Equation Using Cubic B-Spline Galerkin Method" *Fluids* 4, no. 2: 108.
https://doi.org/10.3390/fluids4020108