# Optical Dromions for Spatiotemporal Fractional Nonlinear System in Quantum Mechanics

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### Definition

- If $g\left(t\right)$ = ${t}^{\beta}$, then ${D}_{t}^{\delta}{t}^{\beta}$ = $\frac{\Gamma (\beta +1)}{\Gamma (\beta +1-\alpha )}{t}^{\beta -\delta}$ for $\beta >0$.
- ${D}_{t}^{\delta}\left(g\left(t\right)f\left(t\right)\right)$ = $f\left(t\right){D}_{t}^{\delta}g\left(t\right)+g\left(t\right){D}_{t}^{\delta}f\left(t\right)$.
- ${D}_{t}^{\delta}g\left(f\left(t\right)\right)$ = $\frac{d}{df}g\left(f\left(t\right)\right){D}_{t}^{\delta}f\left(t\right)$ = ${D}_{g}^{\delta}g\left(f\left(t\right)\right){\left(\frac{d}{dt}f\left(t\right)\right)}^{\delta}$.

## 2. Painlevé Test (P-Test)

## 3. Implementation of Painlevé Test (P-Test)

#### 3.1. Dominant Behaviour Calculation

#### 3.2. Calculation for Resonances

#### 3.3. Computation of Integration Constants and Compatibility Condition

## 4. Auxiliary Equation Mapping (AEM) Technique

## 5. Implementation of Auxiliary Equation Mapping (AEM)

**Case 1:**

**Case 2:**

**Case 3:**

**Case 4:**

## 6. Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Kumar, A.; Chauhan, H.V.S.; Ravichandran, C.; Nisar, K.S.; Baleanu, D. Existence of solutions of non-autonomous fractional differential equations with integral impulse condition. Adv. Differ. Equ.
**2020**, 2020, 434. [Google Scholar] [CrossRef] - Shaikh, A.S.; Nisar, K.S. Transmission dynamics of fractional order Typhoid fever model using Caputo–Fabrizio operator. Chaos Solitons Fractals
**2019**, 128, 355–365. [Google Scholar] [CrossRef] - Valliammal, N.; Ravichandran, C. Results on fractional neutral integro-differential systems with state-dependent delay in Banach spaces. Nonlinear Stud.
**2018**, 25, 159–171. [Google Scholar] - Al-Smadi, M.; Arqub, O.A. Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates. Appl. Math. Comput.
**2019**, 342, 280–294. [Google Scholar] [CrossRef] - Al-Smadi, M.; Freihat, A.; Khalil, H.; Momani, S.; Ali Khan, R. Numerical multistep approach for solving fractional partial differential equations. Int. J. Comput. Methods
**2017**, 14, 1750029. [Google Scholar] [CrossRef] - Luchko, Y.; Gorenflo, R. Scale-invariant solutions of a partial differential equation of fractional order. Fract. Calc. Appl. Anal.
**1998**, 1, 63–78. [Google Scholar] - Buckwar, E.; Luchko, Y. Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations. J. Math. Anal. Appl.
**1998**, 227, 81–97. [Google Scholar] [CrossRef] [Green Version] - Xu, Y.; He, Z. The short memory principle for solving Abel differential equation of fractional order. Comput. Math. Appl.
**2011**, 62, 4796–4805. [Google Scholar] [CrossRef] [Green Version] - Veeresha, P.; Prakasha, D.G.; Kumar, S. A fractional model for propagation of classical optical solitons by using nonsingular derivative. Math. Methods Appl. Sci.
**2020**. [Google Scholar] [CrossRef] - Hasan, S.; Al-Smadi, M.; El-Ajou, A.; Momani, S.; Hadid, S.; Al-Zhour, Z. Numerical approach in the Hilbert space to solve a fuzzy Atangana-Baleanu fractional hybrid system. Chaos Solitons Fractals
**2021**, 143, 110506. [Google Scholar] [CrossRef] - Valliammal, N.; Ravichandran, C.; Nisar, K.S. Solutions to fractional neutral delay differential nonlocal systems. Chaos Solitons Fractals
**2020**, 138, 109912. [Google Scholar] [CrossRef] - Kumar, S.; Ghosh, S.; Lotayif, M.S.; Samet, B. A model for describing the velocity of a particle in Brownian motion by Robotnov function based fractional operator. Alex. Eng. J.
**2020**, 59, 1435–1449. [Google Scholar] [CrossRef] - Bekir, A.; Güner, O.; Cevikel, A.C. Fractional complex transform and exp-function methods for fractional differential equations. In Abstract and Applied Analysis; Hindawi: London, UK, 2013. [Google Scholar]
- Yakar, C.; Çiçek, M.; Gücen, M.B. Boundedness and Lagrange stability of fractional order perturbed system related to unperturbed systems with initial time difference in Caputo’s sense. Adv. Differ. Equ.
**2011**, 2011, 54. [Google Scholar] [CrossRef] [Green Version] - Bashiri, T.; Vaezpour, S.M.; Nieto, J.J. Approximating solution of Fabrizio-Caputo Volterra’s model for population growth in a closed system by homotopy analysis method. J. Funct. Spaces
**2018**, 2018, 3152502. [Google Scholar] [CrossRef] [Green Version] - Sene, N.; Abdelmalek, K. Analysis of the fractional diffusion equations described by Atangana-Baleanu-Caputo fractional derivative. Chaos Solitons Fractals
**2019**, 127, 158–164. [Google Scholar] [CrossRef] - Scherer, R.; Kalla, S.L.; Tang, Y.; Huang, J. The Grünwald–Letnikov method for fractional differential equations. Comput. Math. Appl.
**2011**, 62, 902–917. [Google Scholar] [CrossRef] [Green Version] - Aljahdaly, N.H.; Akgül, A.; Shah, R.; Mahariq, I.; Kafle, J. A Comparative Analysis of the Fractional-Order Coupled Korteweg–De Vries Equations with the Mittag–Leffler Law. J. Math.
**2022**, 2022, 8876149. [Google Scholar] [CrossRef] - Javeed, S.; Baleanu, D.; Waheed, A.; Khan, M.S.; Affan, H. Analysis of homotopy perturbation method for solving fractional order differential equations. Mathematics
**2019**, 7, 40. [Google Scholar] [CrossRef] [Green Version] - Odibat, Z.M.; Momani, S. Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul.
**2006**, 7, 27–34. [Google Scholar] [CrossRef] - Agarwal, P.; Karimov, E.; Mamchuev, M.; Ruzhansky, M. On boundary-value problems for a partial differential equation with Caputo and Bessel operators. In Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science; Springer: Berlin, Germany, 2017; pp. 707–718. [Google Scholar]
- Al-Smadi, M.; Arqub, O.A.; Momani, S. Numerical computations of coupled fractional resonant Schrödinger equations arising in quantum mechanics under conformable fractional derivative sense. Phys. Scr.
**2020**, 95, 075218. [Google Scholar] [CrossRef] - Feynman, R.P.; Hibbs, A.R.; Styer, D.F. Quantum Mechanics and Path Integrals; Courier Corporation: North Chelmsford, MA, USA, 2010. [Google Scholar]
- Laskin, N. Fractional Schrödinger equation. Phys. Rev. E
**2012**, 66, 056108. [Google Scholar] [CrossRef] [Green Version] - Laskin, N. Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A
**2000**, 268, 298–305. [Google Scholar] [CrossRef] [Green Version] - Naber, M. Time fractional Schrödinger equation. J. Math. Phys.
**2004**, 45, 3339–3352. [Google Scholar] [CrossRef] [Green Version] - Muslih, S.I.; Agrawal, O.P.; Baleanu, D. A Fractional Schrödinger Equation and Its Solution. Int. J. Theor. Phys.
**2010**, 49, 1746–1752. [Google Scholar] [CrossRef] - Messiah, A. Quantum Mechanics; Courier Corporation: North Chelmsford, MA, USA, 2014. [Google Scholar]
- Guo, B.; Han, Y.; Xin, J. Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation. Appl. Math. Comput.
**2008**, 204, 468–477. [Google Scholar] [CrossRef] - Jumarie, G. An approach to differential geometry of fractional order via modified Riemann-Liouville derivative. Acta Math. Sin. Engl. Ser.
**2019**, 28, 1741–1768. [Google Scholar] [CrossRef] - Jumarie, G. Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions. Appl. Math. Lett.
**2009**, 22, 378–385. [Google Scholar] [CrossRef] - Iomin, A. Fractional-time Schrödinger equation: Fractional dynamics on a comb. Chaos Solitons Fractals
**2011**, 44, 348–352. [Google Scholar] [CrossRef] [Green Version] - Tofighi, A. Probability structure of time fractional Schrödinger equation. Acta Phys. Pol.-Ser. A Gen. Phys.
**2009**, 116, 114. [Google Scholar] [CrossRef] - Alabedalhadi, M. Exact travelling wave solutions for nonlinear system of spatiotemporal fractional quantum mechanics equations. Alex. Eng. J.
**2022**, 61, 1033–1044. [Google Scholar] [CrossRef] - Hulstman, M.V. The Painlevé analysis and exact travelling wave solutions to nonlinear partial differential equations. Math. Comput. Model.
**1993**, 18, 151–156. [Google Scholar] [CrossRef] - Iqbal, M.; Zhang, Y. Painlevé analysis for (2 + 1)-dimensional non-linear Schrödinger equation. Appl. Math.
**2017**, 8, 1539–1545. [Google Scholar] [CrossRef] [Green Version] - Ablowitz, M.J.; Ramani, A.; Segur, H.A. Connection between nonlinear evolution equations and ordinary differential equation of P-type. J. Math. Phys.
**1980**, 21, 715–721. [Google Scholar] [CrossRef] - Weiss, J.; Tabor, M.; Carnevale, G. The Painlevé property for partial differential equations. J. Math. Phys.
**1983**, 24, 522–526. [Google Scholar] [CrossRef] - Ramani, A.; Grammaticos, B.; Bountis, T. The Painlevé property and singularity analysis of integrable and non-integrable systems. Phys. Rep.
**1989**, 180, 159–245. [Google Scholar] [CrossRef] - Baldwin, D.E. Symbolic Algorithms and Software for the Painlevé Test and Recursion Operator for Nonlinear Partial Diffrential Equations. Ph.D. Thesis, Colorado School of Mines, Golden, CO, USA, 2004. [Google Scholar]
- Abdou, M.A. A generlaized auxiliary equation method and its applications. Nonlinear Dyn.
**2008**, 52, 95–102. [Google Scholar] [CrossRef] - Seadawy, A.R.; Cheemaa, N. Application of extended modified auxiliary equation mapping method for high order dispersive extended nonlinear Schrödinger equation in nonlinear optics. Mod. Phys. Lett. B
**2019**, 33, 1950203. [Google Scholar] [CrossRef] - Wu, G.Z.; Yu, L.J.; Wang, Y.Y. Fractional optical solitons of the space-time fractional nonlinear Schrödinger equation. Optik
**2020**, 207, 164405. [Google Scholar] [CrossRef] - Rezazadeh, H.; Odabasi, M.; Tariq, K.U.; Abazari, R.; Baskonus, H.M. On the conformable nonlinear Schrödinger equation with second order spatiotemporal and group velocity dispersion coefficients. Chin. J. Phys.
**2021**, 72, 403–414. [Google Scholar] [CrossRef] - Liang, X.; Khaliq, A.Q.M.; Bhatt, H.; Furati, K.M. The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations. Numer. Algorithms
**2017**, 76, 939–958. [Google Scholar] [CrossRef] - Abdel-Salam, E.A.B.; Yousif, E.A. Solution of nonlinear space-time fractional differential equations using the fractional Riccati expansion method. Math. Probl. Eng.
**2013**, 2013, 846283. [Google Scholar] [CrossRef] - Li, M.; Gu, X.M.; Huang, C.; Fei, M.; Zhang, G. A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations. J. Comput. Phys.
**2018**, 358, 256–282. [Google Scholar] [CrossRef] - Guo, B.; Huang, D. Existence and stability of standing waves for nonlinear fractional Schrödinger equations. J. Math. Phys.
**2012**, 53, 083702. [Google Scholar] [CrossRef]

**Figure 1.**The parameters are $a=2$, $b=1.5$, $c=1$, $k=2$, $s=2$, $m=1$, $n=1$, $\alpha =1$, $\beta =1$, and $\Gamma =1$. The figure provides a pictorial illustration of ${p}_{1,1}(x,t)$: (

**a**) displays the 3D plot in [−0.01, 0.01] and [−2, 2], (

**b**) displays the 2D plot in [−2, 2] and [−5, 5], (

**c**) displays the contour plot.

**Figure 2.**The parameters are $a=2$, $k=2$, $s=2$, $m=1$, $n=1$, $b=1.5$, $c=1$, $\gamma =2$, $\alpha =1$, $\beta =1$, $\Gamma =1$, and $\delta =5$. The figure provides the pictorial illustration of ${q}_{1,2}(x,t)$: (

**a**) displays the 3D plot in [−0.01, 0.01] and [−2, 2], (

**b**) displays the 2D plot in [−20, 20] and [−10, 10], (

**c**) displays the contour plot.

**Figure 3.**The parameters are $a=2$, $k=2$, $s=2$, $m=1$, $b=1.5$, $c=1$, $n=1$, $\gamma =2$, $\alpha =1$, $\beta =1$, and $\Gamma =1$. The figure provides the pictorial illustration of ${r}_{1,3}(x,t)$: (

**a**) displays the 3D plot in [−3, 3] and [−0.03, 0.03], (

**b**) displays the 2D plot in [−3, 3] and [−5, 5], (

**c**) displays the contour plot.

**Figure 4.**The parameters are $a=2$, $k=2$, $s=2$, $m=1$, $n=1$, $b=1.5$, $c=1$, $\gamma =2$, $\alpha =1$, $\beta =1$, and $\Gamma =1$. The figure provides the pictorial illustration of ${q}_{1,4}(x,t)$: (

**a**) displays the 3D plot in [−1, 1] and [−0.01, 0.01], (

**b**) displays the 2D plot in [−10, 10] and [−20, 20], (

**c**) displays the contour plot.

**Figure 5.**The parameters are $a=2$, $s=2$, $m=1$, $n=1$, $\gamma =2$, $\alpha =1$, $b=1.5$, $c=1$, $k=2$, $\beta =1$, and $\Gamma =1$. The figure provides the pictorial illustration of ${p}_{1,5}(x,t)$: (

**a**) displays the 3D plot in [−0.07, 0.07] and [−0.9, 0.9], (

**b**) displays the 2D plot in [−10, 10] and [−20, 20], (

**c**) displays the contour plot.

**Figure 6.**The parameters are $a=2$, $k=2$, $s=2$, $m=1$, $n=1$, $\gamma =2$, $\alpha =1$, $b=1.5$, $c=1$, $\beta =1$, and $\Gamma =1$. The figure provides the pictorial illustration of ${p}_{2,1}(x,t)$: (

**a**) displays the 3D plot in [−3, 3] and [−0.4, 0.4], (

**b**) displays the 2D plot in [−7, 7] and [−2, 2], (

**c**) displays the contour plot.

**Figure 7.**The parameters are $a=2$, $k=2$, $s=2$, $m=1$, $n=1$, $b=1.5$, $c=1$, $\gamma =2$, $\alpha =1$, $\beta =1$, and $\Gamma =1$. The figure provides the pictorial illustration of ${p}_{2,2}(x,t)$: (

**a**) displays the 3D plot in [−3, 3] and [−0.4, 0.4], (

**b**) displays the 2D plot in [−7, 7] and [−2, 2], (

**c**) displays the contour plot.

**Figure 8.**The parameters are $a=2$, $k=2$, $s=2$, $m=1$, $n=1$, $b=1.5$, $c=1$, $\alpha =1$, $\beta =1$, and $\Gamma =1$. Pictorial illustration of ${p}_{2,3}(x,t)$: (

**a**) displays the 3D plot in [−0.01, 0.01] and [−3.75, 3.75], (

**b**) displays the 2D plot in [−5, 5] and [−10, 10], (

**c**) displays the contour plot.

**Figure 9.**The parameters are $a=2$, $k=2$, $s=2$, $m=1$, $n=1$, $b=1.5$, $c=1$, $\alpha =1$, $\beta =1$, and $\Gamma =1$. Pictorial illustration of ${q}_{2,4}(x,t)$: (

**a**) displays the 3D plot in [−1, 1] and [−0.75, 0.75], (

**b**) displays the 2D plot in [−10, 10] and [−17, 17], (

**c**) displays the contour plot.

**Figure 10.**The parameters are $a=2$, $k=2$, $s=2$, $m=1$, $n=1$, $b=1.5$, $c=1$, $\gamma =2$, $\alpha =1$, $\beta =1$, and $\Gamma =1$. The figure provides the pictorial illustration of ${r}_{2,5}(x,t)$: (

**a**) displays the 3D plot in [−0.01, 0.01] and [−0.75, 0.75], (

**b**) displays the 2D plot in [−20, 20] and [−10, 10], (

**c**) displays the contour plot.

**Figure 11.**The parameters are $a=2$, $k=2$, $s=2$, $m=1$, $n=1$, $b=1.5$, $c=1$, $\gamma =2$, $\alpha =1$, $\beta =1$, and $\Gamma =1$. The figure provides the pictorial illustration of ${r}_{3,1}(x,t)$: (

**a**) displays the 3D plot in [−0.01, 0.01] and [−0.75, 0.75], (

**b**) displays the 2D plot in [−5, 5] and [−7, 7], (

**c**) displays the contour plot.

**Figure 12.**The parameters are $a=2$, $k=2$, $s=2$, $m=1$, $b=1.5$, $c=1$, $n=1$, $\gamma =2$, $\alpha =1$, $\beta =1$, and $\Gamma =1$. The figure provides the pictorial illustration of ${q}_{3,2}(x,t)$: (

**a**) displays the 3D plot in [−1, 1] and [−0.03, 0.03], (

**b**) displays the 2D plot in [−5, 5] and [−10, 10], (

**c**) displays the contour plot.

**Figure 13.**The parameters are $a=2$, $k=2$, $s=2$, $m=1$, $n=1$, $b=1.5$, $c=1$, $\alpha =1$, $\beta =1$, and $\Gamma =1$. The figure provides the pictorial illustration of ${p}_{3,3}(x,t)$: (

**a**) displays the 3D plot in [−0.01, 0.01] and [−10, 10], (

**b**) displays the 2D plot in [−5, 5] and [−10, 10], (

**c**) displays the contour plot.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Khoso, I.A.; Katbar, N.M.; Akram, U.
Optical Dromions for Spatiotemporal Fractional Nonlinear System in Quantum Mechanics. *Quantum Rep.* **2023**, *5*, 546-564.
https://doi.org/10.3390/quantum5030036

**AMA Style**

Khoso IA, Katbar NM, Akram U.
Optical Dromions for Spatiotemporal Fractional Nonlinear System in Quantum Mechanics. *Quantum Reports*. 2023; 5(3):546-564.
https://doi.org/10.3390/quantum5030036

**Chicago/Turabian Style**

Khoso, Ihsan A., Nek Muhammad Katbar, and Urooj Akram.
2023. "Optical Dromions for Spatiotemporal Fractional Nonlinear System in Quantum Mechanics" *Quantum Reports* 5, no. 3: 546-564.
https://doi.org/10.3390/quantum5030036