Diverse optical solitons solutions of the fractional complex Ginzburg-Landau equation via two altered methods

Imran Siddique; Khush Bukht Mehdi; Sayed M Eldin; Asim Zafar; Imran Siddique; Khush Bukht Mehdi; Sayed M Eldin; Asim Zafar

doi:10.3934/math.2023581

AIMS Mathematics

2023, Volume 8, Issue 5: 11480-11497. doi: 10.3934/math.2023581

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Research article

Diverse optical solitons solutions of the fractional complex Ginzburg-Landau equation via two altered methods

1.
Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan
2.
Center of Research, Faculty of Engineering, Future University in Egypt New Cairo 11835, Egypt
3.
Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Pakistan

Received: 09 December 2022 Revised: 31 January 2023 Accepted: 06 February 2023 Published: 14 March 2023
MSC : 5C05, 35Q53, 76B25

This work evaluates the fractional complex Ginzburg-Landau equation in the sense of truncated M- fractional derivative and analyzes its soliton solutions and other new solutions in the appearance of a detuning factor in non-linear optics. The multiple, bright, and bright-dark soliton solutions of this equation are obtained using the modified $\left({{{G'} / {{G^2}}}} \right)$ and $\left({{1 / {G'}}} \right) - $expansion methods. The equation is evaluated with Kerr law, quadratic –cubic law and parabolic law non-linear fibers. To shed light on the behavior of solitons, the graphical illustrations in the form of 2D and 3D of the obtained solutions are represented for different values of various parameters. All of the solutions have been verified by substitution into their corresponding equations with the aid of a symbolic software package. The various forms of solutions to the aforementioned nonlinear equation that arises in fluid dynamics and nonlinear processes are presented. Moreover, we guarantee that all the solutions are new and an excellent contribution in the existing literature of solitary wave theory.
Citation: Imran Siddique, Khush Bukht Mehdi, Sayed M Eldin, Asim Zafar. Diverse optical solitons solutions of the fractional complex Ginzburg-Landau equation via two altered methods[J]. AIMS Mathematics, 2023, 8(5): 11480-11497. doi: 10.3934/math.2023581

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Abstract

This work evaluates the fractional complex Ginzburg-Landau equation in the sense of truncated M- fractional derivative and analyzes its soliton solutions and other new solutions in the appearance of a detuning factor in non-linear optics. The multiple, bright, and bright-dark soliton solutions of this equation are obtained using the modified $\left({{{G'} / {{G^2}}}} \right)$ and $\left({{1 / {G'}}} \right) - $expansion methods. The equation is evaluated with Kerr law, quadratic –cubic law and parabolic law non-linear fibers. To shed light on the behavior of solitons, the graphical illustrations in the form of 2D and 3D of the obtained solutions are represented for different values of various parameters. All of the solutions have been verified by substitution into their corresponding equations with the aid of a symbolic software package. The various forms of solutions to the aforementioned nonlinear equation that arises in fluid dynamics and nonlinear processes are presented. Moreover, we guarantee that all the solutions are new and an excellent contribution in the existing literature of solitary wave theory.

References

[1]	L. V. C. Hoan, S. Owyed, M. Inc, L. Ouahid, M. A. Abdou, Y. M.Chu, New explicit optical solitons of fractional nonlinear evolution equation via three different methods, Results Phys., 18 (2020), 103209. https://doi.org/10.1016/j.rinp.2020.103209 doi: 10.1016/j.rinp.2020.103209
[2]	K. Hosseini, K.Sadri, M. Mirzazadeh, Y. M. Chu, A. Ahmadian, B. A. Pansera, et al., A high-order nonlinear Schrödinger equation with the weak non-local nonlinearity and its optical solitons, Results Phys., 23 (2021), 104035. https://doi.org/10.1016/j.rinp.2021.104035 doi: 10.1016/j.rinp.2021.104035
[3]	N. Raza, A. Jhangeer, S. Arshed, A. R. Butt, Y. M. Chu, Dynamical analysis and phase portraits of two-mode waves in different media, Results Phys., 19 (2020), 103650. https://doi.org/10.1016/j.rinp.2020.103650 doi: 10.1016/j.rinp.2020.103650
[4]	Q. Zhang, J. S. Hesthaven, Z. Z Sun, Y. Ren, Pointwise error estimate in difference setting for the two-dimensional nonlinear fractional complex Ginzburg-Landau equation, Adv. Comput. Math., 47 (2021). https://doi.org/10.1007/s10444-021-09862-x doi: 10.1007/s10444-021-09862-x
[5]	Q. Zhang, L. Zhang, H. W. Sun, A three-level finite difference method with preconditioning technique for two-dimensional nonlinear fractional complex Ginzburg–Landau equations, J. Comput. Appl. Math., 389 (2021), 113355. https://doi.org/10.1016/j.cam.2020.113355 doi: 10.1016/j.cam.2020.113355
[6]	H. Rezazadeh, N. Ullah, L. Akinyemi, A. Shah, S. M. M. Alizamin, Y.M.Chu, et al., Optical soliton solutions of the generalized non-autonomous nonlinear Schrödinger equations by the new Kudryashov's method, Results Phys., 24 (2021), 104179. https://doi.org/10.1016/j.rinp.2021.104179 doi: 10.1016/j.rinp.2021.104179
[7]	S. Abbagari, A. Houwe, Y. Saliou, Douvagaï, Y. M. Chu, M. Inc, et al., Analytical survey of the predator–prey model with fractional derivative order, AIP Adv., 11 (2021), 035127. https://doi.org/10.1063/5.0038826 doi: 10.1063/5.0038826
[8]	M. S. Osman, B. Ghanbari, J. A. T. Machado, New complex waves in nonlinear optics based on the complex Ginzburg-Landau equation with Kerr law nonlinearity, Eur. Phys. J. Plus, 134 (2019), 20. https://doi.org/10.1140/epjp/i2019-12442-4 doi: 10.1140/epjp/i2019-12442-4
[9]	Y. M. Chu, M. A. Shallal, S. Mehdi Mirhosseini-Alizamini, H. Rezazadeh, S. Javeed, D. Baleanu, Application of modified extended Tanh technique for solving complex Ginzburg-Landau equation considering Kerr law nonlinearity, CMC Comput. Mater. Con., 66 (2020), 1369–1378.
[10]	W. J. Zhu, Y. H. Xia, Y. Z. Bai, Traveling wave solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity, Appl. Math. Comput., 382 (2020), 125342. https://doi.org/10.1016/j.amc.2020.125342 doi: 10.1016/j.amc.2020.125342
[11]	W. Liu, W. Yu, C. Yang, M. Liu, Y. Zhang, M. Lei, Analytic solutions for the generalized complex Ginzburg-Landau equation in fiber lasers, Nonlinear Dyn., 89 (2017), 2933–2939. https://doi.org/10.1007/s11071-017-3636-5 doi: 10.1007/s11071-017-3636-5
[12]	M. Inc, A. I. Aliyu, A. Yusuf, D. Baleanu, Optical solitons for complex Ginzburg-Landau model in nonlinear optics, Optik, 158 (2018), 368–375.
[13]	A. H. Arnous, A. R. Seadawy, R. T. Alqahtani, A. Biswas, Optical solitons with complex Ginzburg-Landau equation by modified simple equation method, Optik, 144 (2017), 475–480.
[14]	A. H. Khater, D. K. Callebaut, A. R. Seadawy, General soliton solutions of an n-dimensional complex Ginzburg-Landau equation, Phys. Scr., 62 (2000), 353–357. https://doi.org/10.1238/Physica.Regular.062a00353 doi: 10.1238/Physica.Regular.062a00353
[15]	A. Das, A. Biswas, M. Ekici, Q. Zhou, A. S. Alshomrani, M. R. Belic, Optical solitons with complex Ginzburg-Landau equation for two nonlinear forms using F-expansion, Chinese J. Phys., 61 (2019), 255–261. https://doi.org/10.1016/j.cjph.2019.08.009 doi: 10.1016/j.cjph.2019.08.009
[16]	D. S. Oliveira, E. C. de Oliveira, On a Caputo-type fractional derivative, Adv. Pure Appl. Math., 10 (2019), 81–91. https://doi.org/10.1515/apam-2017-0068 doi: 10.1515/apam-2017-0068
[17]	T. M. Atanackovic, S. Pilipovic, D. Zorica, Properties of the Caputo-Fabrizio fractional derivative and its distributional settings, Fract. Calcul. Appl. Anal., 21 (2018), 29–44. https://doi.org/10.1515/fca-2018-0003 doi: 10.1515/fca-2018-0003
[18]	M. D. Ortigueira, Fractional calculus for scientists and engineers, Berlin: Springer, 2011.
[19]	R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
[20]	B. Ghanbari, J. F. Gomez-Aguilar, The generalized exponential rational function method for Radhakrishnan-kundu-lakshmanan equation with Beta conformable time derivative, Revista Mexicana de Fisica, 65 (2019), 503–518. https://doi.org/10.31349/revmexfis.65.503 doi: 10.31349/revmexfis.65.503
[21]	B. A. Jacobs, A new Grunwald-Letnikov derivative derived from a second-order scheme, Abstr. Appl. Anal., 2015 (2015), 952057. https://doi.org/10.1155/2015/952057 doi: 10.1155/2015/952057
[22]	J. Vanterler da C. Sousa, E. Capelas de. Oliveira, A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Int. J. Anal. Appl., 16 (2018), 83–96. https://doi.org/10.48550/arXiv.1704.08187 doi: 10.48550/arXiv.1704.08187
[23]	K. S. Al-Ghafri, Soliton behaviours for the conformable space-time fractional complex Ginzburg-Landau equation in optical fibers, Symmetry, 12 (2020), 219. https://doi.org/10.3390/sym12020219 doi: 10.3390/sym12020219
[24]	C. Huang, Z. Li, New exact solutions of the fractional complex Ginzburg-Landau equation, Math. Problems Eng., 2021 (2021), 6640086. https://doi.org/10.1155/2021/6640086 doi: 10.1155/2021/6640086
[25]	E. Yaşar, Y. Yıldırım, Q. Zhou, S. P. Moshokoa, M. Z. Ullah, H. Triki, et al., Perturbed dark and singular optical solitons in polarization preserving fibers by modified simple equation method, Superlatti. Micro., 111 (2017), 487–498. https://doi.org/10.1016/j.spmi.2017.07.004 doi: 10.1016/j.spmi.2017.07.004
[26]	T. A. Sulaiman, H. M. Baskonus, A. Bulut, Optical solitons and other solutions to the conformable space-time fractional complex Ginzburg-Landau equation under Kerr law nonlinearity, Pramana J. Phys., 58 (2018), 91. https://doi.org/10.1007/s12043-018-1635-9 doi: 10.1007/s12043-018-1635-9
[27]	M. A. Abdou, A. A. Soliman, A. Biswas, M. Ekici, Q. Zhou, S. P. Moshokoa, Dark-singular combo optical solitons with fractional complex Ginzburg-Landau equation, Optik, 171 (2018), 463–467. https://doi.org/10.1016/j.ijleo.2018.06.076 doi: 10.1016/j.ijleo.2018.06.076
[28]	S. Arshed, Soliton solutions of fractional complex Ginzburg-Landau equation with Kerr law and non-Kerr law media, Optik, 160 (2018), 322–332. https://doi.org/10.1016/j.ijleo.2018.02.022 doi: 10.1016/j.ijleo.2018.02.022
[29]	B. Ghanbari, J. F. G`o.an-Aguilar, Optical soliton solutions of the Ginzburg-Landau equation with conformable derivative and Kerr law nonlinearity, Revista Mexicana de Fisica, 65 (2019), 73–81.
[30]	P. H. Lu, B. H. Wang, C. Q. Dai, Fractional traveling wave solutions of the (2+1)-dimensional fractional complex Ginzburg-Landau equation via two methods, Math. Methods Appl. Sci., 43 (2020), 8518–8526. https://doi.org/10.1002/mma.6511 doi: 10.1002/mma.6511
[31]	A. Hussain, A. Jhangeer, Optical solitons of fractional complex Ginzburg-Landau with conformable, beta, and M-truncated derivatives: a comparative study, Adv. Differ. Equ., 2020 (2020), 612. https://doi.org/10.1186/s13662-020-03052-7 doi: 10.1186/s13662-020-03052-7
[32]	G. Akram, M. Sadaf, H. Mariyam, A comparative study of the optical solitons for the fractional complex Ginzburg-Landau equation using different fractional differential operators, Optik, 256 (2022), 168626. https://doi.org/10.1016/j.ijleo.2022.168626 doi: 10.1016/j.ijleo.2022.168626
[33]	M. Sadaf, G. Akram, M. Dawood, An investigation of fractional complex Ginzburg-Landau equation with Kerr law nonlinearity in the sense of conformable, beta and M‑truncated derivatives, Opt. Quant. Electron., 54 (2022), 248. https://doi.org/10.1007/s11082-022-03570-6 doi: 10.1007/s11082-022-03570-6
[34]	A. Zafar, M. Shakeel, Optical solitons of nonlinear complex Ginzburg-Landau equation via two modified expansion schemes, Opt. Quant. Electron., 54 (2022), 5. https://doi.org/10.1007/s11082-021-03393-x doi: 10.1007/s11082-021-03393-x
[35]	H. M. Baskonus, J. F. Gómez-Aguilar, New singular soliton solutions to the longitudinal wave equation in a magneto-electro-elastic circular rod with M-derivative, Mod. Phys. Lett. B, 33 (2019), 1950251. https://doi.org/10.1142/S0217984919502518 doi: 10.1142/S0217984919502518
[36]	B. Ghanbari, J. F. Gómez-Aguilar, New exact optical soliton solutions for nonlinear Schrödinger equation with second-order spatio-temporal dispersion involving M-derivative, Mod. Phys. Lett. B, 33 (2019), 1950235. https://doi.org/10.1142/S021798491950235X doi: 10.1142/S021798491950235X
[37]	S. Demiray, O. Unsal, A. Bekir, New exact solutions for Boussinesq type equations by using $\left({{{G'} / {G, {1 / G}}}} \right)$ and $\left({{1 / {G'}}} \right)$expansion method, Acta Phys. Pol. A, 125 (2014), 1093–1098. https://doi.org/10.12693/APhysPolA.125.1093 doi: 10.12693/APhysPolA.125.1093
[38]	Y. Zhang, L. Zhang, J. Pang, Application of $\left({{{G'} / {{G.2}}}} \right)$ expansion method for solving Schrodinger's equation with three-order dispersion, Adv. Appl. Math., 6 (2017), 212–217. https://doi.org/10.12677/aam.2017.62024 doi: 10.12677/aam.2017.62024
[39]	I. Siddique, M. M. M. Jaradat, A. Zafar, K. B. Mehdi, M. S. Osman, Exact traveling wave solutions for two prolific conformable M-Fractional differential equations via three diverse approaches, Results Phys., 28 (2021), 104557. https://doi.org/10.1016/j.rinp.2021.104557 doi: 10.1016/j.rinp.2021.104557
[40]	I. Siddique, K. B. Mehdi, M. M. M. Jaradat, A. Zafar, M. E. Elbrolosy, A. A. Elmandouh, et al., Bifurcation of some new traveling wave solutions for the time-space M-fractional MEW equation via three altered methods, Results Phys., 41 (2022), 105896. https://doi.org/10.1016/j.rinp.2022.105896 doi: 10.1016/j.rinp.2022.105896
[41]	A. Biswas, S. Konar, E. Zerrad, Soliton-soliton interaction with parabolic law nonlinearity, J. Electromag. Waves Appl., 20 (2002), 927–939. https://doi.org/10.1163/156939306776149833 doi: 10.1163/156939306776149833
[42]	E. Yaşar, Y. Yıldırım, Q. Zhou, S. P. Moshokoa, M. Z. Ullah, H. Triki, et al., Perturbed dark and singular optical solitons in polarization preserving fibers by modified simple equation method, Superlatti. Micro., 111 (2017), 487–498. https://doi.org/10.1016/j.spmi.2017.07.004 doi: 10.1016/j.spmi.2017.07.004
[43]	A. Biswas, S. Konar, E. Zerrad, Soliton-soliton interaction with parabolic law nonlinearity, J. Electromag. Waves Appl., 20 (2006), 927–939. https://doi.org/10.1163/156939306776149833 doi: 10.1163/156939306776149833

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