Chaotic behavior and construction of a variety of wave structures related to a new form of generalized q-Deformed sinh-Gordon model using couple of integration norms

Wedad Albalawi; Nauman Raza; Saima Arshed; Muhammad Farman; Kottakkaran Sooppy Nisar; Abdel-Haleem Abdel-Aty; Wedad Albalawi; Nauman Raza; Saima Arshed; Muhammad Farman; Kottakkaran Sooppy Nisar; Abdel-Haleem Abdel-Aty

doi:10.3934/math.2024466

AIMS Mathematics

2024, Volume 9, Issue 4: 9536-9555. doi: 10.3934/math.2024466

Previous Article Next Article

Research article Special Issues

Chaotic behavior and construction of a variety of wave structures related to a new form of generalized q-Deformed sinh-Gordon model using couple of integration norms

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan
3.
Department of Computer Science and Mathematics, Lebanese American University, 1107-2020, Beirut, Lebanon
4.
Faculty of Arts and Science, Department of Mathematics, Near East University, Cyprus, Turkey
5.
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia
6.
Department of Physics, College of Sciences, University of Bisha, Bisha 61922, Saudi Arabia

Received: 16 January 2024 Revised: 19 February 2024 Accepted: 29 February 2024 Published: 07 March 2024
MSC : 35B32, 35C08

The generalized q-deformed sinh Gordon equation (GDSGE) serves as a significant nonlinear partial differential equation with profound applications in physics. This study investigates the GDSGE's mathematical and physical properties, examining its solutions and clarifying the essence of the q-deformation parameter. The Sardar sub-equation method (SSEM) and sine-Gordon expansion method (SGEM) are employed to solve this GDSGE. The synergistic application of these techniques improves our knowledge of the GDSGE and provides a thorough foundation for investigating different evolution models arising in various branches of mathematics and physics. A positive aspect of the proposed methods is that they offer a wide variety of solitons, including bright, singular, dark, combination dark-singular, combined dark-bright, and periodic singular solitons. Obtained solutions demonstrate the method's high degree of reliability, simplicity, and functionalization for various nonlinear equations. To better describe the physical characterization of solutions, a few 2D and 3D visualizations are generated by taking precise values for parameters using mathematical software, Mathematica.
- generalized q-deformed sinh Gordon equation,
- Sardar sub-equation method,
- sine-Gordon expansion method,
- solitons
Citation: Wedad Albalawi, Nauman Raza, Saima Arshed, Muhammad Farman, Kottakkaran Sooppy Nisar, Abdel-Haleem Abdel-Aty. Chaotic behavior and construction of a variety of wave structures related to a new form of generalized q-Deformed sinh-Gordon model using couple of integration norms[J]. AIMS Mathematics, 2024, 9(4): 9536-9555. doi: 10.3934/math.2024466

Related Papers:

Abstract

The generalized q-deformed sinh Gordon equation (GDSGE) serves as a significant nonlinear partial differential equation with profound applications in physics. This study investigates the GDSGE's mathematical and physical properties, examining its solutions and clarifying the essence of the q-deformation parameter. The Sardar sub-equation method (SSEM) and sine-Gordon expansion method (SGEM) are employed to solve this GDSGE. The synergistic application of these techniques improves our knowledge of the GDSGE and provides a thorough foundation for investigating different evolution models arising in various branches of mathematics and physics. A positive aspect of the proposed methods is that they offer a wide variety of solitons, including bright, singular, dark, combination dark-singular, combined dark-bright, and periodic singular solitons. Obtained solutions demonstrate the method's high degree of reliability, simplicity, and functionalization for various nonlinear equations. To better describe the physical characterization of solutions, a few 2D and 3D visualizations are generated by taking precise values for parameters using mathematical software, Mathematica.

References

[1]	N. Raza, A. R. Seadawy, M. Kaplan, A. R. Butt, Symbolic computation and sensitivity analysis of nonlinear Kudryashov's dynamical equation with applications, Phys. Scr., 96 (2021), 105216. https://doi.org/10.1088/1402-4896/ac0f93 doi: 10.1088/1402-4896/ac0f93
[2]	J. H. He, A new approach to nonlinear partial differential equations, Commun. Nonlinear Sci. Numer. Simul., 2 (1997), 230–235. https://doi.org/10.1016/S1007-5704(97)90007-1 doi: 10.1016/S1007-5704(97)90007-1
[3]	M. M. Khater, S. Muhammad, A. Al-Ghamdi, M. Higazy, Novel soliton wave solutions of the Vakhnenko-Parkes equation arising in the relaxation medium, J. Ocean Eng. Sci., 2022, In Press. https://doi.org/10.1016/j.joes.2022.02.015
[4]	A. V. Buryak, P. Di Trapani, D. V. Skryabin, S. Trillo, Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications, Phys. Rep., 370 (2002), 63–235. https://doi.org/10.1016/S0370-1573(02)00196-5 doi: 10.1016/S0370-1573(02)00196-5
[5]	S. Singla, N. S. Saini, Higher-order dust kinetic Alfvén wave solitons and quasi-periodic waves in a polarized dusty plasma, Waves Random Complex Media, 2023. https://doi.org/10.1080/17455030.2023.2238067
[6]	N. Raza, F. Salman, A. R. Butt, M. L. Gandarias, Lie symmetry analysis, soliton solutions and qualitative analysis concerning to the generalized q-deformed sinh-Gordon equation, Commun. Nonlinear Sci. Numer. Simul., 116 (2023), 106824. https://doi.org/10.1016/j.cnsns.2022.106824 doi: 10.1016/j.cnsns.2022.106824
[7]	B. Silindir, Soliton solutions of q-Toda lattice by Hirota direct method, Adv. Differ. Equ., 2012 (2012), 1–22. https://doi.org/10.1186/1687-1847-2012-121 doi: 10.1186/1687-1847-2012-121
[8]	O. Billet, M. Joye, The Jacobi model of an elliptic curve and side-channel analysis, In: Applied algebra, algebraic algorithms, and error-correcting codes, 2003, 34–42. https://doi.org/10.1007/3-540-44828-4_5
[9]	Y. Fang, G. Z. Wu, Y. Y. Wang, C. Q. Dai, Data-driven femtosecond optical soliton excitations and parameters discovery of the high-order NLSE using the PINN, Nonlinear Dyn., 105 (2021), 603–616. https://doi.org/10.1007/s11071-021-06550-9 doi: 10.1007/s11071-021-06550-9
[10]	R. J. Kuo, M. R. Setiawan, T. P. Q. Nguyen, Sequential clustering and classification using deep learning technique and multi-objective sine-cosine algorithm, Comput. Ind. Eng., 173 (2022), 108695. https://doi.org/10.1016/j.cie.2022.108695 doi: 10.1016/j.cie.2022.108695
[11]	A. Mahmood, H. M. Srivastava, M. Abbas, F. A. Abdullah, P. O. Mohammed, D. Baleanu, et al., Optical soliton solutions of the coupled Radhakrishnan-Kundu-Lakshmanan equation by using the extended direct algebraic approach, Heliyon, 9 (2023), e20852. https://doi.org/10.1016/j.heliyon.2023.e20852 doi: 10.1016/j.heliyon.2023.e20852
[12]	N. Raza, M. Abdullah, A. R. Butt, Analytical soliton solutions of Biswas-Milovic equation in Kerr and non-Kerr law media, Optik, 157 (2018), 993–1002. https://doi.org/10.1016/j.ijleo.2017.11.043 doi: 10.1016/j.ijleo.2017.11.043
[13]	N. Raza, S. Arshed, A. Javid, Optical solitons and stability analysis for the generalized second-order nonlinear Schrödinger equation in an optical fiber, Int. J. Nonlinear Sci. Numer. Simul., 21 (2020), 855–863. https://doi.org/10.1515/ijnsns-2019-0287 doi: 10.1515/ijnsns-2019-0287
[14]	N. Raza, A. Batool, M. Inc, New hyperbolic and rational form solutions of (2+1)-dimensional generalized Korteweg-de Vries model, J. Ocean Eng. Sci., 2022, In Press. https://doi.org/10.1016/j.joes.2022.04.021
[15]	K. K. Ali, M. F. Alotaibi, M. Omri, M. S. Mehanna, A. H. Abdel-Aty, Some traveling wave solutions to the fifth-order nonlinear wave equation using three techniques: Bernoulli sub-ODE, modified auxiliary equation, and ($G^{'}/G$)-expansion methods, J. Math., 2023 (2023), 1–22. https://doi.org/10.1155/2023/7063620 doi: 10.1155/2023/7063620
[16]	J. Pan, M. U. Rahman, Rafiullah, Breather-like, singular, periodic, interaction of singular and periodic solitons, and a-periodic solitons of third-order nonlinear Schrödinger equation with an efficient algorithm, Eur. Phys. J. Plus, 138 (2023), 912. https://doi.org/10.1140/epjp/s13360-023-04530-z doi: 10.1140/epjp/s13360-023-04530-z
[17]	H. Rezazadeh, A. Zabihi, A. G. Davodi, R. Ansari, H. Ahmad, S. W. Yao, New optical solitons of double sine-Gordon equation using exact solutions methods, Results Phys., 49 (2023), 106452. https://doi.org/10.1016/j.rinp.2023.106452 doi: 10.1016/j.rinp.2023.106452
[18]	B. Kemaloalu, G. Yel, H. Bulut, An application of the rational sine-Gordon method to the Hirota equation, Opt. Quant. Electron., 55 (2023), 658. https://doi.org/10.1007/s11082-023-04930-6
[19]	D. Bahns, N. Pinamonti, K. Rejzner, Equilibrium states for the massive sine-Gordon theory in the Lorentzian signature, J. Math. Anal. Appl., 526 (2023), 127249. https://doi.org/10.1016/j.jmaa.2023.127249
[20]	H. U. Rehman, R. Akber, A. M. Wazwaz, H. M. Alshehri, M. S. Osman, Analysis of Brownian motion in stochastic Schrödinger wave equation using Sardar sub-equation method, Optik, 289 (2023), 171305. https://doi.org/10.1016/j.ijleo.2023.171305
[21]	T. Rasool, R. Hussain, M. A. Al Sharif, W. Mahmoud, M. S. Osman, A variety of optical soliton solutions for the M-truncated paraxial wave equation using Sardar-subequation technique, Opt. Quant. Electron., 55 (2023), 396. https://doi.org/10.1007/s11082-023-04655-6
[22]	N. Ullah, M. I. Asjad, A. Hussanan, A. Akgül, W. R. Alharbi, H. Algarni, et al., Novel wave structures for two nonlinear partial differential equations arising in the nonlinear optics via Sardar-subequation method, Alex. Eng. J., 71 (2023), 105–113. https://doi.org/10.1016/j.aej.2023.03.023
[23]	Z. Li, C. Y. Liu, Chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic law nonlinearity and spatio-temporal dispersion, Results Phys., 56 (2024), 107305. https://doi.org/10.1016/j.rinp.2023.107305
[24]	R. F. Luo, Rafiullah, H. Emadifar, M. U. Rahman, Bifurcations, chaotic dynamics, sensitivity analysis and some novel optical solitons of the perturbed non-linear Schrödinger equation with Kerr law non-linearity, Results Phys., 54 (2023), 107133. https://doi.org/10.1016/j.rinp.2023.107133
[25]	M. Vivas-Cortez, N. Raza, S. S. Kazmi, Y. Chahlaoui, G. A. Basendwah, A novel investigation of dynamical behavior to describe nonlinear wave motion in (3+1)-dimensions, Results Phys., 55 (2023), 107131. https://doi.org/10.1016/j.rinp.2023.107131
[26]	M. H. Rafiq, N. Raza, A. Jhangeer, Dynamic study of bifurcation, chaotic behavior and multi-soliton profiles for the system of shallow water wave equations with their stability, Chaos Solitons Fract., 171 (2023), 113436. https://doi.org/10.1016/j.chaos.2023.113436
[27]	N. Raza, A. Jaradat, G. A. Basendwah, A. Batool, M. M. M. Jaradat, Dynamic analysis and derivation of new optical soliton solutions for the modified complex Ginzburg-Landau model in communication, Alex. Eng. J., 90 (2024), 197–207. https://doi.org/10.1016/j.aej.2024.01.059
[28]	Y. S. Özkan, A study on the solutions of (3+1) conformal time derivative generalized q-deformed sinh-Gordon equation, Celal Bayar Univ. J. Sci., 19 (2023), 219–229. https://doi.org/10.18466/cbayarfbe.1264314
[29]	L. D. Faddeev, Modular double of a quantum group, Math. Phys. Stud., 21 (2000), 149–156. https://doi.org/10.48550/arXiv.math/9912078
[30]	H. M. Srivastava, Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis, Iran. J. Sci. Technol. Trans. A Sci., 44 (2020), 327–344. https://doi.org/10.1007/s40995-019-00815-0
[31]	M. A. Ali, H. Budak, A. Akkurt, Y. M. Chu, Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus, Open Math., 19 (2021), 440–449. https://doi.org/10.1515/math-2021-0020
[32]	H. M. Srivastava, M. K. Aouf, A. O. Mostafa, Some properties of analytic functions associated with fractional q-calculus operators, Miskolc Math. Notes, 20 (2019), 1245–1260. https://doi.org/10.18514/MMN.2019.3046
[33]	H. Eleuch, Some analytical solitary wave solutions for the generalized q-deformed sinh-Gordon equation: $\frac{\partial^{2}\theta}{\partial z \partial \xi} = \alpha\left[\sinh_q(\beta \theta^{\gamma})\right]^p-\delta$, Adv. Math. Phys., 2018 (2018), 1–7. https://doi.org/10.1155/2018/5242757 doi: 10.1155/2018/5242757
[34]	K. K. Ali, N. Al-Harbi, A. H. Abdel-Aty, Traveling wave solutions to (3+1) conformal time derivative generalized q-deformed sinh-Gordon equation, Alex. Eng. J., 65 (2023), 233–243. https://doi.org/10.1016/j.aej.2022.10.020 doi: 10.1016/j.aej.2022.10.020
[35]	K. K. Ali, Analytical and numerical study for the generalized q-deformed sinh-Gordon equation, Nonlinear Eng., 12 (2023), 20220255. https://doi.org/10.1515/nleng-2022-0255 doi: 10.1515/nleng-2022-0255
[36]	S. S. Kazmi, A. Jhangeer, N. Raza, H. I. Alrebdi, A. H. Abdel-Aty, H. Eleuch, The analysis of bifurcation, quasi-periodic and solitons patterns to the new form of the generalized q-deformed sinh-Gordon equation, Symmetry, 15 (2023), 1–19. https://doi.org/10.3390/sym15071324 doi: 10.3390/sym15071324
[37]	L. Q. Bai, J. M. Qi, Y. Q. Sun, Further physical study about solution structures for nonlinear q-deformed sinh-Gordon equation along with bifurcation and chaotic behaviors, Nonlinear Dyn., 111 (2023), 20165–20199. https://doi.org/10.1007/s11071-023-08882-0 doi: 10.1007/s11071-023-08882-0
[38]	K. K. Ali, A. H. Abdel-Aty, H. Eleuch, New soliton solutions for the conformal time derivative q-deformed physical model, Results Phys., 42 (2022), 105993. https://doi.org/10.1016/j.rinp.2022.105993 doi: 10.1016/j.rinp.2022.105993
[39]	N. Raza, S. Arshed, H. I. Alrebdi, A. H. Abdel-Aty, H. Eleuch, Abundant new optical soliton solutions related to q-deformed sinh-Gordon model using two innovative integration architectures, Results Phys., 35 (2022), 105358. https://doi.org/10.1016/j.rinp.2022.105358 doi: 10.1016/j.rinp.2022.105358

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)