Analytical discovery of dark soliton lattices in (2+1)-dimensional generalized fractional Kundu-Mukherjee-Naskar equation

Abdulah A. Alghamdi; Abdulah A. Alghamdi

doi:10.3934/math.20241123

AIMS Mathematics

2024, Volume 9, Issue 8: 23100-23127. doi: 10.3934/math.20241123

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Analytical discovery of dark soliton lattices in (2+1)-dimensional generalized fractional Kundu-Mukherjee-Naskar equation

Abdulah A. Alghamdi ^{1,2
,
,}

1.
Mathematical Modeling and Applied Computation (MMAC) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah-21589, Saudi Arabia
2.
Center of Modern Mathematical Sciences and their Applications (CMMSA), King Abdulaziz, University, Jeddah-21589, Saudi Arabia

Received: 10 June 2024 Revised: 07 July 2024 Accepted: 15 July 2024 Published: 29 July 2024

This research explored optical soliton solutions for the (2+1)-dimensional generalized fractional Kundu-Mukherjee-Naskar equation (gFKMNE), which is a nonlinear model for explaining pulse transmission in communication structures and optical fibers. Two enhanced variants of $ (\frac{G'}{G}) $-expansion method were employed, namely, extended $ (\frac{G'}{G}) $-expansion method and the generalized $ (r+\frac{G'}{G}) $-expansion method, based on the wave transformation of the model into integer-order nonlinear ordinary differential equations (NODEs). By assuming a series-form solution for the resultant NODEs, these strategic methods further translated them into a system of nonlinear algebraic equations. Solving these equations provided optical soliton solutions for gFKMNE using the Maple-13 tool. Through 3D and contour visuals, it was revealed that the constructed soliton solutions are periodically arranged in the optical medium, forming dark soliton lattices. These dark soliton lattices are significant in several domains, such as optical signal processing, optical communications, and nonlinear optics.
Citation: Abdulah A. Alghamdi. Analytical discovery of dark soliton lattices in (2+1)-dimensional generalized fractional Kundu-Mukherjee-Naskar equation[J]. AIMS Mathematics, 2024, 9(8): 23100-23127. doi: 10.3934/math.20241123

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Abstract

This research explored optical soliton solutions for the (2+1)-dimensional generalized fractional Kundu-Mukherjee-Naskar equation (gFKMNE), which is a nonlinear model for explaining pulse transmission in communication structures and optical fibers. Two enhanced variants of $ (\frac{G'}{G}) $-expansion method were employed, namely, extended $ (\frac{G'}{G}) $-expansion method and the generalized $ (r+\frac{G'}{G}) $-expansion method, based on the wave transformation of the model into integer-order nonlinear ordinary differential equations (NODEs). By assuming a series-form solution for the resultant NODEs, these strategic methods further translated them into a system of nonlinear algebraic equations. Solving these equations provided optical soliton solutions for gFKMNE using the Maple-13 tool. Through 3D and contour visuals, it was revealed that the constructed soliton solutions are periodically arranged in the optical medium, forming dark soliton lattices. These dark soliton lattices are significant in several domains, such as optical signal processing, optical communications, and nonlinear optics.

References

[1]	S. Phoosree, S. Payakkarak, W. Thadee, Physical Impact of the Nonlinear Space and Time Fractional Fluid Dynamic Equation, Physical Impact of the Nonlinear Space and Time Fractional Fluid Dynamic Equation, 2022.
[2]	H. Yasmin, A. S. Alshehry, A. H. Ganie, A. M. Mahnashi, R. Shah. Perturbed Gerdjikov-Ivanov equation: Soliton solutions via Backlund transformation, Optik, 298 (2024), 171576.
[3]	A. Seadawy, A. Sayed, Soliton solutions of cubic-quintic nonlinear SchrÃ¶dinger and variant Boussinesq equations by the first integral method, Filomat, 31 (2017), 4199–4208.
[4]	D. Baleanu, Y. Karaca, L. Vaizquez, J. E. Macaas-Daaz, Advanced fractional calculus, differential equations and neural networks: Analysis, modeling and numerical computations, Phys. Scripta, 98 (2023), 110201.
[5]	H. Almusawa, A. Jhangeer, A study of the soliton solutions with an intrinsic fractional discrete nonlinear electrical transmission line, Fractal Fract., 6 (2022), 334.
[6]	B. P. Moghaddam, Z. S. Mostaghim, A novel matrix approach to fractional finite difference for solving models based on nonlinear fractional delay differential equations, Ain Shams Eng. J., 5 (2014), 585–594.
[7]	C. Zhu, M. Al-Dossari, S. Rezapour, S. A. M. Alsallami, B. Gunay, Bifurcations, chaotic behavior, and optical solutions for the complex Ginzburg-Landau equation, Results Phys., 59 (2024), 107601. https://doi.org/10.1016/j.rinp.2024.107601 doi: 10.1016/j.rinp.2024.107601
[8]	C. Zhu, M. Al-Dossari, S. Rezapour, S. Shateyi, B. Gunay, Analytical optical solutions to the nonlinear Zakharov system via logarithmic transformation, Results Phys., 56 (2024), 107298. https://doi.org/10.1016/j.rinp.2023.107298 doi: 10.1016/j.rinp.2023.107298
[9]	C. Zhu, S. A. O. Abdallah, S. Rezapour, S. Shateyi, On new diverse variety analytical optical soliton solutions to the perturbed nonlinear Schrodinger equation, Results Phys., 54 (2023), 107046. https://doi.org/10.1016/j.rinp.2023.107046 doi: 10.1016/j.rinp.2023.107046
[10]	Z. Pan, J. Pan, L. Sang, Z. Ding, M. Liu, L. Fu, X. Wan, Highly efficient solution-processable four-coordinate boron compound: A thermally activated delayed fluorescence emitter with short-live]d phosphorescence for OLEDs with small efficiency roll-off, Chem. Eng. J., 483 (2024), 149221. https://doi.org/10.1016/j.cej.2024.149221 doi: 10.1016/j.cej.2024.149221
[11]	L. Liu, S. Zhang, L. Zhang, G. Pan, J. Yu, Multi-UUV Maneuvering Counter-Game for Dynamic Target Scenario Based on Fractional-Order Recurrent Neural Network, IEEE Trans. Cyber., 53 (2023), 4015–4028. https://doi.org/10.1109/TCYB.2022.3225106 doi: 10.1109/TCYB.2022.3225106
[12]	Y. Kai, J. Ji, Z. Yin, Study of the generalization of regularized long-wave equation, Nonlinear Dyn., 107 (2022), 2745–2752. 10.1007/s11071-021-07115-6 doi: 10.1007/s11071-021-07115-6
[13]	N. Raza, M. S. Osman, A. H. Abdel-Aty, S. Abdel-Khalek, H. R. Besbes, Optical solitons of space-time fractional Fokas-Lenells equation with two versatile integration architectures, Adv. Differ. Equa., (2020), 517.
[14]	H. M. Baskonus, H. Bulut, T. A. Sulaiman, New Complex Hyperbolic Structures to the Lonngren-Wave Equation by Using Sine-Gordon Expansion Method, Appl. Math. Nonlinear Sci., 4 (2019), 129–138.
[15]	S. Jiong, Auxiliary equation method for solving nonlinear partial differential equations, Phys. Lett. A, 309 (2003), 387–396.
[16]	M. S. Hashemi, M. EINAS, M. Bayram, Symmetry properties and exact solutions of the time fractional Kolmogo-rov-Petrovskii- Piskunov equation, Rev. Mex. Fis., 65 (2019), 529–535.
[17]	L. Akinyemi, M. Mirzazadeh, S. Amin Badri, K. Hosseini. Dynamical solitons for the perturbated Biswas-Milovic equation with Kudryashovaes law of refractive index using the first integral method, J. Mod. Opt., 69 (2022), 172–182.
[18]	K. Hosseini, R. Ansari. New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method, Waves Random Complex Media 27 (2017), 628-636.
[19]	D. Kumar, M. Kaplan, New analytical solutions of (2+1)-dimensional conformable time fractional Zoomeron equation via two distinct techniques, Chin. J. Phys., 56 (2018), 2173–2185.
[20]	M. M. Khater, D. Lu, R. A. Attia, Dispersive long wave of nonlinear fractionalWu-Zhang system via a modified auxiliary equation method, AIP Adv., 9 (2019), 25003.
[21]	B. Zhang, W. Zhu, Y. Xia, Y. Bai, A Unified Analysis of Exact TravelingWave Solutions for the Fractional-Order and Integer-Order Biswas-Milovic Equation: Via Bifurcation Theory of Dynamical System, Qual. Theory Dyn. Syst., 19 (2020), 11.
[22]	H. Khan, S. Barak, P. Kumam, M. Arif, Analytical solutions of fractional Klein-Gordon and gas dynamics equations, via the (G'/G)-expansion method, Symmetry, 11 (2019), 566.
[23]	R. Ali, E. Tag-eldin, A comparative analysis of generalized and extended (G'/G)-Expansion methods for travelling wave solutions of fractional Maccari's system with complex structure, Alexandria Eng. J., 79 (2023), 508–530.
[24]	H. Khan, R. Shah, J. F. Gómez-Aguilar, D. Baleanu, P. Kumam, Travelling waves solution for fractional-order biological population model, Math. Modell. Nat. Phenom., 16 (2021), 32.
[25]	F. Wang, S. A. Salama, M. M. Khater, Optical wave solutions of perturbed time-fractional nonlinear Schrodinger equation, J. Ocean Eng. Sci., 2022.
[26]	M. M. Khater, Physics of crystal lattices and plasma, analytical and numerical simulations of the Gilson-Pickering equation, Results Phys, 44 (2023), 106193.
[27]	M. M. Bhatti, D. Q. Lu, An application of Nwoguaes Boussinesq model to analyze the head-on collision process between hydroelastic solitary waves, Open Phys, 17 (2019), 177–191. https://doi.org/10.1515/phys-2019-0018
[28]	M. M. Al-Sawalha, S. Noor, S. Alshammari, A. H. Ganie, A. Shafee, Analytical insights into solitary wave solutions of the fractional Estevez-Mansfield-Clarkson equation, AIMS Mathematics, 9 (2024), 13589–13606. http://doi.org/10.3934/math.2024663 doi: 10.3934/math.2024663
[29]	J. H. He, X. H. Wu, Exp-function method for nonlinear wave equations, Chaos, Soliton. Fract., 30 (2006), 3, 700-708.
[30]	A. Iftikhar, A. Ghafoor, T. Zubair, S. Firdous, S. T. Mohyud-Din, solutions of (2+ 1) dimensional generalized KdV, Sin Gordon and Landau-Ginzburg-Higgs Equations, Sci. Res. Essays, 8 (2013), 1349–1359.
[31]	J. F. Alzaidy, Fractional sub-equation method and its applications to the space-time fractional differential equations in mathematical physics, Brit. J. Math. Comput. Sci., 3 (2013), 153–163.
[32]	K. R. Raslan, K. K. Ali, M. A. Shallal, The modified extended tanh method with the Riccati equation for solving the space-time fractional EW and MEW equations, Chaos, Soliton. Fract., 103 (2017), 404–409.
[33]	M. Alqhtani, K. M. Saad, R. Shah, W. M. Hamanah, Discovering novel soliton solutions for (3+ 1)-modified fractional Zakharov-Kuznetsov equation in electrical engineering through an analytical approach, Opt. Quant. Electron., 55 (2023), 1149.
[34]	H. Yasmin, N. H. Aljahdaly, A. M. Saeed, R. Shah. Investigating Families of Soliton Solutions for the Complex Structured Coupled Fractional Biswas-Arshed Model in Birefringent Fibers Using a Novel Analytical Technique, Fractal Fract., 7 (2023), 491.
[35]	M. M. Al-Sawalha, H. Yasmin, R. Shah, A. H. Ganie, K. Moaddy, Unraveling the Dynamics of Singular Stochastic Solitons in Stochastic Fractional Kuramoto-Sivashinsky Equation, Fractal Fract., 7 (2023), 753.
[36]	M. Aldandani, A. A. Altherwi, M. M. Abushaega, Propagation patterns of dromion and other solitons in nonlinear Phi-Four ($\phi^4$) equation, AIMS Mathematics, 9 (2024), 19786–19811. https://doi.org/10.3934/math.2024966 doi: 10.3934/math.2024966
[37]	M. Cinar, A. Secer, M. Ozisik, M. Bayram, Derivation of optical solitons of dimensionless Fokas-Lenells equation with perturbation term using Sardar sub-equation method, Opt. Quant. Electron., 54 (2022), 402.
[38]	B. Ghanbari, On novel non differentiable exact solutions to local fractional Gardneraes equation using an effective technique, Math. Methods Appl. Sci., 44 (2021), 4673–4685.
[39]	M. M. Al-Sawalha, A. Khan, O. Y. Ababneh, T. Botmart, Fractional view analysis of Kersten-Krasil'shchik coupled KdV-mKdV systems with non-singular kernel derivatives, AIMS Mathematics, 7 (2022), 18334–18359. http://doi.org/10.3934/math.20221010 doi: 10.3934/math.20221010
[40]	S. Alshammari, M. M. Al-Sawalha, Approximate analytical methods for a fractional-order nonlinear system of Jaulent-Miodek equation with energy-dependent Schrodinger potential, Fractal Fract., 7 (2023), 140.
[41]	A. A. Alderremy, N. Iqbal, S. Aly, K.Nonlaopon, Fractional series solution construction for nonlinear fractional reaction-diffusion Brusselator model utilizing Laplace residual power series, Symmetry, 14 (2022), 1944.
[42]	R. Ali, S. Barak, A. Altalbe, Analytical study of soliton dynamics in the realm of fractional extended shallow water wave equations, Phys. Scripta, 99 (2024), 065235.
[43]	Z. Avazzadeh, O. Nikan, J. A. T. Machado, Solitary wave solutions of the generalized Rosenau-KdV-RLW equation, Mathematics, 8 (2020), 1601.
[44]	O. Nikan, Z. Avazzadeh, M. N. Rasoulizadeh, Soliton wave solutions of nonlinear mathematical models in elastic rods and bistable surfaces, Eng. Anal. Bound. Elem., 143 (2022), 14–27.
[45]	A. Kundu, A. Mukherjee, T. Naskar, Modeling rogue waves through exact dynamical lamps soliton controlled by ocean currents, Proc. R. Soc. A., 470 (2014), 20130576.
[46]	M. Ekici, A. Sonmezoglu, A. Biswas, M. Belic, Optical solitons in (2+ 1)-dimensions with Kundu-Mukherjee-Naskar equation by extended trial function scheme, Chin. J. Phys., 57 (2019), 72—77.
[47]	Y. Yildirim, Optical solitons to Kundu-Mukherjee-Naskar model in birefringent fibers with trial equation approach, Optik, 183 (2019), 1026–1031.
[48]	Y. Yildirim, Optical solitons to Kundu-Mukherjee-Naskar model with modified simple equation approach, Optik, 184 (2019), 247–252.
[49]	Y. Tang, Traveling wave optical solutions for the generalized fractional kundu-mukherjee-naskar (gfkmn) model, Mathematics, 11 (2023), 2583.
[50]	H. Ganerhan, F. S. Khodadad, H. Rezazadeh, M. M. Khater, Exact optical solutions of the (2+ 1) dimensions Kundu-Mukherjee-Naskar model via the new extended direct algebraic method, Modern Phys. Lett. B, 34 (2020), 2050225.
[51]	S. T. R. Rizvi, I. Afzal, K. Ali, Dark and singular optical solitons for Kundu-Mukherjee-Naskar model, Modern Phys. Lett. B, 34 (2020), 2050074.
[52]	R. A. Talarposhti, P. Jalili, H. Rezazadeh, B. Jalili, D. D. Ganji, W. Adel, A. Bekir, Optical soliton solutions to the (2+ 1)-dimensional Kundu-Mukherjee-Naskar equation, Int. J. Modern Phys. B, 34 (2020), 2050102.
[53]	I. Onder, A. Secer, M. Ozisik, M. Bayram, On the optical soliton solutions of Kundu-Mukherjee-Naskar equation via two different analytical methods, Optik, 257 (2022), 168761.
[54]	A. Zafar, M. Raheel, K. K. Ali, M. Inc, A. Qaisar, Optical solitons to the Kundu-Mukherjee-Naskar equation in (2+ 1)-dimensional form via two analytical techniques, J. Laser Appl., 34 (2022), 2.
[55]	D. Kumar, G. C. Paul, T. Biswas, A. R. Seadawy, R. Baowali, M. Kamal, H. Rezazadeh, Optical solutions to the Kundu-Mukherjee-Naskar equation: mathematical and graphical analysis with oblique wave propagation, Phys. Scripta, 96 (2020), 025218.
[56]	J. H. He, Variational principle and periodic solution of the Kundu-Mukherjee-Naskar equation, Results Phys., 17 (2020), 103031.
[57]	M. Ekici, A. Sonmezoglu, A. Biswas, M. R. Belic, Optical solitons in (2+ 1)-dimensions with Kundu-Mukherjee-Naskar equation by extended trial function scheme, Chin. J. Phys., 57 (2019), 72–77.
[58]	T. A. Sulaiman H. Bulut, The new extended rational SGEEM for construction of optical solitons to the (2+ 1)-dimensional Kundu-Mukherjee-Naskar model, Appl. Math. Nonlinear Sci., 4 (2019), 513–522.
[59]	K. J. Wang, H. W. Zhu, Periodic wave solution of the Kundu-Mukherjee-Naskar equation in birefringent fibers via the Hamiltonian-based algorithm, Europhys. Lett., 139 (2022), 35002.
[60]	O. González-Gaxiola, A. Biswas, M. Asma, A. K. Alzahrani, Optical dromions and domain walls with the Kundu-Mukherjee-Naskar equation by the Laplace-Adomian decomposition scheme, Regular Chaotic Dyn., 25 (2020), 338–348.
[61]	E. M. Elsayed, K. Nonlaopon, The Analysis of the Fractional-Order Navier-Stokes Equations by a Novel Approach, J. Funct. Spaces, 2022 (2022), 8979447.
[62]	M. Alqhtani, K. M. Saad, W. Weera, W. M. Hamanah, Analysis of the fractional-order local Poisson equation in fractal porous media, Symmetry, 14 (2022), 1323.
[63]	M. Naeem, H. Rezazadeh, A. A. Khammash, S. Zaland, Analysis of the Fuzzy Fractional-Order Solitary Wave Solutions for the KdV Equation in the Sense of Caputo-Fabrizio Derivative, J. Math., 2022 (2022), 3688916.
[64]	M. Naeem, O. F. Azhar, A. M. Zidan, K. Nonlaopon, Numerical Analysis of Fractional-Order Parabolic Equations via Elzaki Transform, J. Funct. Spaces, 2021 (2021), 3484482.
[65]	P. Sunthrayuth, A. M. Zidan, S.Khan, J. Kafle, The Analysis of Fractional-Order Navier-Stokes Model Arising in the Unsteady Flow of a Viscous Fluid via Shehu Transform, J. Funct. Spaces, 2021 (2021), 1029196.
[66]	Z. Hui, A. Wu, D. Han, T. Li, L. Li, J. Gong, et al., Switchable Single- to Multiwavelength Conventional Soliton and Bound-State Soliton Generated from a NbTe2 Saturable Absorber-Based Passive Mode-Locked Erbium-Doped Fiber Laser, ACS Appl. Mater. Interfaces, 16 (2024), 22344–22360. https://doi.org/10.1021/acsami.3c19323 doi: 10.1021/acsami.3c19323
[67]	C. Zhu, M. Al-Dossari, S. Rezapour, B. Gunay, On the exact soliton solutions and different wave structures to the (2+1) dimensional Chaffee-Infante equation, Results Phys., 57 (2024), 107431. https://doi.org/10.1016/j.rinp.2024.107431 doi: 10.1016/j.rinp.2024.107431
[68]	Y. Kai, Z. Yin, Linear structure and soliton molecules of Sharma-Tasso-Olver-Burgers equation, Phys. Lett. A, 452 (2022), 128430. https://doi.org/10.1016/j.physleta.2022.128430 doi: 10.1016/j.physleta.2022.128430
[69]	V. E. Tarasov, On chain rule for fractional derivatives, Commun. Nonlinear Sci. Numer. Simul., 30 (2016), 1–4.
[70]	J. H. He, S. K. Elagan, Z. B. Li, Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A, 376 (2012), 257–259.
[71]	M. Z. Sarikaya, H. Budak, H. Usta, On generalized the conformable fractional calculus, TWMS J. Appl. Eng. Math., 9 (2019), 792–799.

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