Highly dispersive gap solitons for conformable fractional model in optical fibers with dispersive reflectivity solutions using the modified extended direct algebraic method

Mahmoud Soliman; Hamdy M. Ahmed; Niveen Badra; Taher A. Nofal; Islam Samir; Mahmoud Soliman; Hamdy M. Ahmed; Niveen Badra; Taher A. Nofal; Islam Samir

doi:10.3934/math.20241229

AIMS Mathematics

2024, Volume 9, Issue 9: 25205-25222. doi: 10.3934/math.20241229

Previous Article Next Article

Research article Special Issues

Highly dispersive gap solitons for conformable fractional model in optical fibers with dispersive reflectivity solutions using the modified extended direct algebraic method

1.
Department of Physics and Mathematics Engineering, Faculty of Engineering, Ain Shams University, Cairo, Egypt
2.
Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El Shorouk Academy, Cairo, Egypt
3.
Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 19 July 2024 Revised: 16 August 2024 Accepted: 19 August 2024 Published: 28 August 2024
MSC : 26A33, 35C07, 35C08, 35C09

We investigated the dynamics of highly dispersive nonlinear gap solitons in optical fibers with dispersive reflectivity, utilizing a conformable fractional derivative model. The modified extended direct algebraic method was employed to obtain various soliton solutions, including bright solitons and singular solitons, as well as hyperbolic and trigonometric solutions. The key findings demonstrated that the fractional derivative parameter ($ \alpha $) can effectively control the wave propagation, causing a shift in the wave signal while maintaining the same amplitude. This is a novel contribution, as the ability to control soliton properties through the conformable derivative is explored for the first time in this work. The results showcase the significant influence of fractional derivatives in shaping the characteristics of the soliton solutions, which is crucial for accurately modeling the dispersive and nonlocal effects in optical fibers. This research provides insights into the potential applications of fractional calculus in the design and optimization of photonic devices for optical communication systems.
- highly dispersive gap solitons,
- conformable fractional model,
- modified extended direct algebraic method
Citation: Mahmoud Soliman, Hamdy M. Ahmed, Niveen Badra, Taher A. Nofal, Islam Samir. Highly dispersive gap solitons for conformable fractional model in optical fibers with dispersive reflectivity solutions using the modified extended direct algebraic method[J]. AIMS Mathematics, 2024, 9(9): 25205-25222. doi: 10.3934/math.20241229

Related Papers:

Abstract

We investigated the dynamics of highly dispersive nonlinear gap solitons in optical fibers with dispersive reflectivity, utilizing a conformable fractional derivative model. The modified extended direct algebraic method was employed to obtain various soliton solutions, including bright solitons and singular solitons, as well as hyperbolic and trigonometric solutions. The key findings demonstrated that the fractional derivative parameter ($ \alpha $) can effectively control the wave propagation, causing a shift in the wave signal while maintaining the same amplitude. This is a novel contribution, as the ability to control soliton properties through the conformable derivative is explored for the first time in this work. The results showcase the significant influence of fractional derivatives in shaping the characteristics of the soliton solutions, which is crucial for accurately modeling the dispersive and nonlocal effects in optical fibers. This research provides insights into the potential applications of fractional calculus in the design and optimization of photonic devices for optical communication systems.

References

[1]	W. B. Rabie, H. M. Ahmed, A. R. Seadawy, A. Althobaiti, The higher-order nonlinear Schrödinger's dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity via dispersive analytical soliton wave solutions, Opt. Quant. Electron., 53 (2021), 6868. http://doi.org/10.1007/s11082-021-03278-z doi: 10.1007/s11082-021-03278-z
[2]	I. Samir, H. M. Ahmed, M. Mirzazadeh, H. Triki, Derivation of new solitons and other solutions for higher order Sasa–Satsuma equation by using the improved modified extended tanh scheme, Optik, 274 (2023), 170592. http://doi.org/10.1016/j.ijleo.2023.170592 doi: 10.1016/j.ijleo.2023.170592
[3]	Y.-L. Ma, B.-Q. Li, The dynamics on soliton molecules and soliton bifurcation for an extended generalization of Vakhnenko equation, Qual. Theory Dyn. Syst., 23 (2024), 137. http://doi.org/10.1007/s12346-024-01002-2 doi: 10.1007/s12346-024-01002-2
[4]	B.-Q. Li, A.-M. Wazwaz, Y.-L. Ma, Soliton resonances, soliton molecules to breathers, semi-elastic collisions and soliton bifurcation for a multi-component Maccari system in optical fiber, Opt. Quant. Electron., 56 (2024), 573. http://doi.org/10.1007/s11082-023-06224-3 doi: 10.1007/s11082-023-06224-3
[5]	Y.-L. Ma, B.-Q. Li, Interaction behaviors between solitons, breathers and their hybrid forms for a short pulse equation, Qual. Theory Dyn. Syst., 22 (2023), 146. http://doi.org/10.1007/s12346-023-00844-6 doi: 10.1007/s12346-023-00844-6
[6]	Y.-L. Ma, B.-Q. Li, Optical soliton resonances, soliton molecules to breathers for a defocusing Lakshmanan–Porsezian–Daniel system, Opt. Quant. Electron., 56 (2024), 151. http://doi.org/10.1007/s11082-023-05687-8 doi: 10.1007/s11082-023-05687-8
[7]	Y.-L. Ma, B.-Q. Li, Higher-order hybrid rogue wave and breather interaction dynamics for the AB system in two-layer fluids, Math. Comput. Simul., 221, 489–502. https://doi.org/10.1016/j.matcom.2024.03.017
[8]	Y.-L. Ma, B.-Q. Li, Soliton interactions, soliton bifurcations and molecules, breather molecules, breather-to-soliton transitions, and conservation laws for a nonlinear (3+1)-dimensional shallow water wave equation, Nonlinear Dyn., 112 (2024), 2851–2867. http://doi.org/10.1007/s11071-023-09185-0 doi: 10.1007/s11071-023-09185-0
[9]	C. Guo, B. Guo, The existence of global solutions for the fourth-order nonlinear Schrödinger equations, J. Appl. Anal. Comput., 9 (2019), 1183–1192. http://doi.org/10.11948/2156-907X.20190095 doi: 10.11948/2156-907X.20190095
[10]	R. Kohl, A. Biswas, D. Milovic, E. Zerrad, Adiabatic dynamics of Gaussian and super-Gaussian solitons in dispersion-managed optical fibers, Prog. Electromagn. Res., 84 (2008), 27–53. http://doi.org/10.2528/PIER08052703 doi: 10.2528/PIER08052703
[11]	A.-M. Wazwaz, W. Alhejaili, S. A. El-Tantawy, Bright and dark envelope optical solitons for a (2+1)-dimensional cubic nonlinear Schrödinger equation, Optik, 265 (2022), 169525. http://doi.org/10.1016/j.ijleo.2022.169525 doi: 10.1016/j.ijleo.2022.169525
[12]	A.-M. Wazwaz, W. Alhejaili, A. O. AL-Ghamdi, S. A. El-Tantawy, Bright and dark modulated optical solitons for a (2+1)-dimensional optical Schrödinger system with third-order dispersion and nonlinearity, Optik, 274 (2023), 170582. http://doi.org/10.1016/j.ijleo.2023.170582 doi: 10.1016/j.ijleo.2023.170582
[13]	K. K. Ahmed, N. M. Badra, H. M. Ahmed, W. B. Rabie, Soliton solutions and other solutions for Kundu–Eckhaus equation with quintic nonlinearity and Raman effect using the improved modified extended tanh-function method, Mathematics, 10 (2022), 4203. http://doi.org/10.3390/math10224203 doi: 10.3390/math10224203
[14]	Y. Alhojilan, H. M. Ahmed, Novel analytical solutions of stochastic Ginzburg-Landau equation driven by Wiener process via the improved modified extended tanh function method, Alexandria Eng. J., 72 (2023), 269–274. http://doi.org/10.1016/j.aej.2023.02.016 doi: 10.1016/j.aej.2023.02.016
[15]	W. B. Rabie, H. M. Ahmed, Construction of cubic-quartic solitons in optical metamaterials for the perturbed twin-core couplers with Kudryashov's sextic power law using extended F-expansion method, Chaos, Soliton. Fract., 160 (2022), 112289. http://doi.org/10.1016/j.chaos.2022.112289 doi: 10.1016/j.chaos.2022.112289
[16]	T. A. Khalil, N. Badra, H. M. Ahmed, W. B. Rabie, Bright solitons for twin-core couplers and multiple-core couplers having polynomial law of nonlinearity using Jacobi elliptic function expansion method, Alexandria Eng. J., 61 (2022), 11925–11934. https://doi.org/10.1016/j.aej.2022.05.042 doi: 10.1016/j.aej.2022.05.042
[17]	M. Soliman, H. M. Ahmed, N. Badra, I. Samir, Effects of fractional derivative on fiber optical solitons of (2+1) perturbed nonlinear Schrödinger equation using improved modified extended tanh-function method, Opt. Quant. Electron., 56 (2024), 777. https://doi.org/10.1007/s11082-024-06593-3 doi: 10.1007/s11082-024-06593-3
[18]	F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, Discrete solitons in optics, Phys. Rep., 463 (2008), 1–126. https://doi.org/10.1016/j.physrep.2008.04.004 doi: 10.1016/j.physrep.2008.04.004
[19]	M. Soliman, H. M. Ahmed, N. Badra, I. Samir, Dispersive perturbations of solitons for conformable fractional complex Ginzburg–Landau equation with polynomial law of nonlinearity using improved modified extended tanh-function method, Opt. Quant. Electron., 56 (2024), 1084. http://doi.org/10.1007/s11082-024-07018-x doi: 10.1007/s11082-024-07018-x
[20]	M. A. Arefin, U. Sadiya, M. Inc, M. H. Uddin, Adequate soliton solutions to the space–time fractional telegraph equation and modified third-order KdV equation through a reliable technique, Opt. Quant. Electron., 54 (2022), 309. https://doi.org/10.1007/s11082-022-03640-9 doi: 10.1007/s11082-022-03640-9
[21]	U. H. M. Zaman, M. A. Arefin, M. A. Akbar, M. H. Uddin, Explore dynamical soliton propagation to the fractional order nonlinear evolution equation in optical fiber systems, Opt. Quant. Electron., 55 (2023), 1295. https://doi.org/10.1007/s11082-023-05474-5 doi: 10.1007/s11082-023-05474-5
[22]	M. A. Arefin, U. H. M. Zaman, M. H. Uddin, M. Inc, Consistent travelling wave characteristic of space–time fractional modified Benjamin–Bona–Mahony and the space-time fractional Duffing models, Opt. Quant. Electron., 56 (2024), 588. https://doi.org/10.1007/s11082-023-06260-z doi: 10.1007/s11082-023-06260-z
[23]	U. H. M. Zaman, M. A. Arefin, M. A. Akbar, M. H. Uddin, Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique, PLOS ONE, 18 (2023), e0285178. http://doi.org/10.1371/journal.pone.0285178 doi: 10.1371/journal.pone.0285178
[24]	M. A. Arefin, M. A. Khatun, M. S. Islam, M. A. Akbar, M. H. Uddin, Explicit soliton solutions to the fractional order nonlinear models through the Atangana beta derivative, Int. J. Theor. Phys., 62 (2023), 134. https://doi.org/10.1007/s10773-023-05400-1 doi: 10.1007/s10773-023-05400-1
[25]	A. Podder, M. A. Arefin, M. A. Akbar, M. H. Uddin, A study of the wave dynamics of the space-time fractional nonlinear evolution equations of beta derivative using the improved Bernoulli sub-equation function approach, Sci. Rep., 13 (2023), 20478. https://doi.org/10.1038/s41598-023-45423-6 doi: 10.1038/s41598-023-45423-6
[26]	E. M. E. Zayed, M. E. M. Alngar, R. Shohib, A. Biswas, Y. Yıldırım, C. M. Balanica Dragomir, Highly dispersive gap solitons in optical fibers with dispersive reflectivity having parabolic-nonlocal nonlinearity, Ukr. J. Phys. Opt., 25 (2024), 01033–01044.
[27]	W. A. Faridi, M. Iqbal, M. B. Riaz, S. A. AlQahtani, A.-M. Wazwaz, The fractional soliton solutions of dynamical systems arising in plasma physics: A comparative analysis, Alexandria Eng. J., 95 (2024), 247–261. https://doi.org/10.1016/j.aej.2024.03.061 doi: 10.1016/j.aej.2024.03.061
[28]	A. Kajouni, A. Chafiki, K. Hilal, M. Oukessou, A new conformable fractional derivative and applications, Int. J. Differ. Eq., 2021 (2021), 6245435. http://doi.org/10.1155/2021/6245435 doi: 10.1155/2021/6245435
[29]	M. B. Hubert, G. Betchewe, M. Justin, S. Y. Doka, K. T. Crepin, A. Biswas, et al., Optical solitons with Lakshmanan–Porsezian–Daniel model by modified extended direct algebraic method, Optik, 162 (2018), 228–236. https://doi.org/10.1016/j.ijleo.2018.02.091 doi: 10.1016/j.ijleo.2018.02.091
[30]	M. H. Ali, H. M. El-Owaidy, H. M. Ahmed, A. A. El-Deeb, I. Samir, Optical solitons and complexitons for generalized Schrödinger–Hirota model by the modified extended direct algebraic method, Opt. Quant. Electron., 55 (2023), 675. https://doi.org/10.1007/s11082-023-04962-y doi: 10.1007/s11082-023-04962-y
[31]	M. Bilal, J. Iqbal, R. Ali, F. A. Awwad, E. A. A. Ismail, Exploring families of solitary wave solutions for the fractional coupled Higgs system using modified extended direct algebraic method, Fractal Fract., 7 (2023), 653. http://doi.org/10.3390/fractalfract7090653 doi: 10.3390/fractalfract7090653
[32]	I. Samir, N. Badra, H. M. Ahmed, A. H. Arnous, Solitons dynamics in optical metamaterial with quadratic–cubic nonlinearity using modified extended direct algebraic method, Optik, 243 (2021), 166851. http://doi.org/10.1016/j.ijleo.2021.166851 doi: 10.1016/j.ijleo.2021.166851
[33]	O. El-Shamy, R. El-Barkouky, H. M. Ahmed, W. Abbas, I. Samir, Extraction of solitons in magneto–optic waveguides for coupled NLSEs with Kudryashov's law of nonlinearity via modified extended direct algebraic method, Ain Shams Eng. J., 15 (2024), 102477. https://doi.org/10.1016/j.asej.2023.102477 doi: 10.1016/j.asej.2023.102477

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)