Advanced study of nonlinear partial differential equations in ion-acoustic and light pulses: solitons, chaos, and energy perspectives

Syed T. R. Rizvi; M. A. Abdelkawy; Hanadi Zahed; Aly R. Seadawy; Syed T. R. Rizvi; M. A. Abdelkawy; Hanadi Zahed; Aly R. Seadawy

doi:10.3934/math.2025535

AIMS Mathematics

2025, Volume 10, Issue 5: 11842-11879. doi: 10.3934/math.2025535

Previous Article Next Article

Research article Special Issues

Advanced study of nonlinear partial differential equations in ion-acoustic and light pulses: solitons, chaos, and energy perspectives

1.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus 54000, Pakistan
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
3.
Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah 41411, Saudi Arabia

Received: 22 February 2025 Revised: 28 April 2025 Accepted: 06 May 2025 Published: 22 May 2025
MSC : 35J05, 35J10, 35K05, 35L05

The emphasis of this work is to apply the complete discriminant system (CDS) of a polynomial method (CDSPM) to obtain soliton solutions for the extended Korteweg–de Vries equation (EKdVE). The model of the KdV equation is suitable for describing water waves with shallow water, where the water depth is up to the wavelength. We explore exact and analytical solutions such as the hyperbolic function (HF), the Jacobian elliptic function (JEF), solitary wave (SW) solutions, and trigonometric solutions. Additionally, we apply bifurcation concept in qualitative model of the EKdVE. The procedure involves transforming the system into a planer dynamical system through a given transformation and examining bifurcation analysis. We also explore phase portraits which assist in determining the stability of the equilibrium point of the system. Additionally, chaotic behaviour (CB) exhibits the exponential divergence of the orbits, i.e., tiny differences in the initial condition result in widely divergent orbits over a time period. To explore the potential of CB, we insert a perturbed term to the dynamical system and proceed with caution to explore the model. From these, we obtain a linear combination resulting in a strong dynamic mathematical structure. Finally, we apply Jupyter as a machine learning program as well as Mathematica to graph some obtained solutions in various dimensions such as three-dimensional (3D) and two-dimensional (2D) plots by using some packages of PYTHON such as numpy, scipy.integrate, and matplotlib.pyplot. Additionally, we explore energy balance method (EBM), presenting approximate periodic solution of non-linear oscillatory systems. The method is based on principle of conservation of energy, total energy (sum of kinetic energy as well as potential energy) is conserved.
- soliton pulses,
- chaos analysis,
- qualitative analysis
Citation: Syed T. R. Rizvi, M. A. Abdelkawy, Hanadi Zahed, Aly R. Seadawy. Advanced study of nonlinear partial differential equations in ion-acoustic and light pulses: solitons, chaos, and energy perspectives[J]. AIMS Mathematics, 2025, 10(5): 11842-11879. doi: 10.3934/math.2025535

Related Papers:

Abstract

The emphasis of this work is to apply the complete discriminant system (CDS) of a polynomial method (CDSPM) to obtain soliton solutions for the extended Korteweg–de Vries equation (EKdVE). The model of the KdV equation is suitable for describing water waves with shallow water, where the water depth is up to the wavelength. We explore exact and analytical solutions such as the hyperbolic function (HF), the Jacobian elliptic function (JEF), solitary wave (SW) solutions, and trigonometric solutions. Additionally, we apply bifurcation concept in qualitative model of the EKdVE. The procedure involves transforming the system into a planer dynamical system through a given transformation and examining bifurcation analysis. We also explore phase portraits which assist in determining the stability of the equilibrium point of the system. Additionally, chaotic behaviour (CB) exhibits the exponential divergence of the orbits, i.e., tiny differences in the initial condition result in widely divergent orbits over a time period. To explore the potential of CB, we insert a perturbed term to the dynamical system and proceed with caution to explore the model. From these, we obtain a linear combination resulting in a strong dynamic mathematical structure. Finally, we apply Jupyter as a machine learning program as well as Mathematica to graph some obtained solutions in various dimensions such as three-dimensional (3D) and two-dimensional (2D) plots by using some packages of PYTHON such as numpy, scipy.integrate, and matplotlib.pyplot. Additionally, we explore energy balance method (EBM), presenting approximate periodic solution of non-linear oscillatory systems. The method is based on principle of conservation of energy, total energy (sum of kinetic energy as well as potential energy) is conserved.

References

[1]	N. Vassena, F. Avram, R. Adenane, Finding bifurcations in mathematical epidemiology via reaction network methods, Chaos, 34 (2024), 123163. https://doi.org/10.1063/5.0221082 doi: 10.1063/5.0221082
[2]	J. B. Li, Y. Q. Yang, W. Y. Sun, Breather wave solutions on the Weierstrass elliptic periodic background for the (2+1)-dimensional generalized variable-coefficient KdV equation, Chaos, 34 (2024), 023141. https://doi.org/10.1063/5.0192185 doi: 10.1063/5.0192185
[3]	Y.-H. Huang, R. Guo, Wave breaking, dispersive shock wave, and phase shift for the defocusing complex modified KdV equation, Chaos, 34 (2024), 103103. https://doi.org/10.1063/5.0231741 doi: 10.1063/5.0231741
[4]	Z. J. Lin, Z. Y. Yan, Interactions and asymptotic analysis of $N$-soliton solutions for the $n$-component generalized higher-order Sasa-Satsuma equations, Chaos, 34 (2024), 123145. https://doi.org/10.1063/5.0237425 doi: 10.1063/5.0237425
[5]	L. L. Zhang, H. T. Han, $N$-Soliton solutions of coupled Schrödinger-Boussinesq equation with variable coefficients, Commun. Nonlinear Sci., 138 (2024), 108185. https://doi.org/10.1016/j.cnsns.2024.108185 doi: 10.1016/j.cnsns.2024.108185
[6]	Y. G. Kao, C. H. Wang, H. W. Xia, Y. Cao, Projective synchronization for uncertain fractional reaction-diffusion systems via adaptive sliding mode control based on finite-time scheme, In: Analysis and control for fractional-order systems, Singapore: Springer, 2024,141–163. https://doi.org/10.1007/978-981-99-6054-5_8
[7]	F. Y. Liu, Y. Q. Yang, Q. Chang, Synchronization of fractional-order delayed neural networks with reaction–diffusion terms: distributed delayed impulsive control, Commun. Nonlinear Sci., 124 (2023), 107303. https://doi.org/10.1016/j.cnsns.2023.107303 doi: 10.1016/j.cnsns.2023.107303
[8]	X. Meng, Y. G. Kao, H. R. Karimi, C. C. Gao, Global Mittag-Leffler stability for fractional-order coupled systems on network without strong connectedness, Sci. China Inf. Sci., 63 (2020), 132201. https://doi.org/10.1007/s11432-019-9946-6 doi: 10.1007/s11432-019-9946-6
[9]	Y. Cao, Y. G. Kao, Z. Wang, X. S. Yang, J. H. Park, W. Xie, Sliding mode control for uncertain fractional-order reaction–diffusion memristor neural networks with time delays, Neural Networks, 178 (2024), 106402. https://doi.org/10.1016/j.neunet.2024.106402 doi: 10.1016/j.neunet.2024.106402
[10]	Y. G. Kao, C. H. Wang, H. W. Xia, Y. Cao, Global Mittag-Leffler synchronization of coupled delayed fractional reaction-diffusion Cohen-Grossberg neural networks via sliding mode control, In: Analysis and control for fractional-order systems, Singapore: Springer, 2024,121–140. https://doi.org/10.1007/978-981-99-6054-5_7
[11]	A. Ali, A. R. Seadawy, D. C. Lu, New solitary wave solutions of some nonlinear models and their applications, Adv. Differ. Equ., 2018 (2018), 232. https://doi.org/10.1186/s13662-018-1687-7 doi: 10.1186/s13662-018-1687-7
[12]	M. S. Ullah, M. Z. Ali, H.-O. Roshid. Stability analysis, $\phi^6$ model expansion method, and diverse chaos-detecting tools for the DSKP model, Sci. Rep., 15 (2025), 13658. https://doi.org/10.1038/s41598-025-98275-7 doi: 10.1038/s41598-025-98275-7
[13]	A. R. Seadawy, D. C. Lu, C. Yue, Travelling wave solutions of the generalized nonlinear fifth-orderKdV water wave equations and its stability, J. Taibah Univ. Sci., 11 (2017), 623–633. https://doi.org/10.1016/j.jtusci.2016.06.002 doi: 10.1016/j.jtusci.2016.06.002
[14]	M. Arshad, A. R. Seadawy, D. C. Lu, Elliptic function and solitary wave solutions of the higher-order nonlinear Schrödinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability, Eur. Phys. J. Plus, 132 (2017), 371. https://doi.org/10.1140/epjp/i2017-11655-9 doi: 10.1140/epjp/i2017-11655-9
[15]	A. H. Khater, D. K. Callebaut, W. Malfliet, A. R. Seadawy, Nonlinear dispersive Rayleigh-Taylor instabilities in magnetohydrodynamic flows, Phys. Scr., 64 (2001), 533. https://doi.org/10.1238/Physica.Regular.064a00533 doi: 10.1238/Physica.Regular.064a00533
[16]	L. Akinyemi, A. Houwe, S. Abbagari, A.-M. Wazwaz, H. M. Alshehri, M. S. Osman, Effects of the higher-order dispersion on solitary waves and modulation instability in a monomode fiber, Optik, 288 (2023), 171202. https://doi.org/10.1016/j.ijleo.2023.171202 doi: 10.1016/j.ijleo.2023.171202
[17]	F. Badshah, K. U. Tariq, M. Inc, M. Aslam, M. Zeeshan, On the study of bright, dark and optical wave structures for the coupled fractional nonlinear Schrödinger equations in plasma physics, Opt. Quant. Electron., 55 (2023), 1170. https://doi.org/10.1007/s11082-023-05434-z doi: 10.1007/s11082-023-05434-z
[18]	M. Inc, R. T. Alqahtani, M. S. Iqbal, Statblity analysis and consistent solitary wave solutions for the reaction diffusion regularised nonlinear model, Results Phys., 54 (2023), 107053. https://doi.org/10.1016/j.rinp.2023.107053 doi: 10.1016/j.rinp.2023.107053
[19]	U. K. Manda, A. Das, W.-X. Ma, Integrability, breather, rogue wave, lump, lump-multi-stripe, and lump-multi-soliton solutions of a (3+1)-dimensional nonlinear evolution equation, Phys. Fluids, 36 (2024), 037151. https://doi.org/10.1063/5.0195378 doi: 10.1063/5.0195378
[20]	A.-M. Wazwaz, W. Alhejaili, S. A. El-Tantawy, Study on extensions of (modified) Korteweg–de Vries equations: Painlevé integrability and multiple soliton solutions in fluid mediums, Phys. Fluids, 35 (2023), 093110. https://doi.org/10.1063/5.0169733 doi: 10.1063/5.0169733
[21]	A. Neirameh, M. Eslami, New travelling wave solutions for plasma model of extended K-dV equation, Afr. Mat., 30 (2019), 335–344. https://doi.org/10.1007/s13370-018-00651-2 doi: 10.1007/s13370-018-00651-2
[22]	A. Neirameh, N. Memarian, New analytical soliton type solutions for double layers structure model of extended KDV equation, Computational Methods for Differential Equations, 5 (2017), 256–270.
[23]	M. Eslami, Optical solutions to a conformable fractional extended KdV model equation, Partial Differential Equations in Applied Mathematics, 8 (2023), 100562. https://doi.org/10.1016/j.padiff.2023.100562 doi: 10.1016/j.padiff.2023.100562
[24]	L.-C. Shi, Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations, Comput. Phys. Commun., 181 (2010), 317–324. https://doi.org/10.1016/j.cpc.2009.10.006 doi: 10.1016/j.cpc.2009.10.006
[25]	T. Usman, I. Hossain, M. S. Ullah, M. M. Hasan, Soliton, multistability, and chaotic dynamics of the higher-order nonlinear Schrödinger equation, Chaos, 35 (2025), 043141. https://doi.org/10.1063/5.0266469 doi: 10.1063/5.0266469
[26]	S. T. R. Rizvi, S. Shabbir, Optical soliton solution via complete discrimination system approach along with bifurcation and sensitivity analyses for the Gerjikov-Ivanov equation, Optik, 294 (2023), 171456. https://doi.org/10.1016/j.ijleo.2023.171456 doi: 10.1016/j.ijleo.2023.171456
[27]	Y. Kai, S. Q. Chen, B. L. Zheng, K. Zhang, N. Yang, W. L. Xu, Qualitative and quantitative analysis of nonlinear dynamics by the complete discrimination system for polynomial method, Chaos Soliton. Fract., 141 (2020), 110314. https://doi.org/10.1016/j.chaos.2020.110314 doi: 10.1016/j.chaos.2020.110314
[28]	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), 1324. https://doi.org/10.3390/sym15071324 doi: 10.3390/sym15071324
[29]	S. Wael, E. A. Ahmed, A. R. Seadawy, R. S. Ibrahim, Bifurcation, similarity reduction, conservation laws and exact solutions of modified-Korteweg-de Vries–Burger equation, Opt. Quant. Electron., 55 (2023), 262. https://doi.org/10.1007/s11082-022-04517-7 doi: 10.1007/s11082-022-04517-7
[30]	A. H. Tedjani, A. R. Seadawy, S. T. R. Rizvi, E. Solouma, Construction of Hamiltonina and optical solitons along with bifurcation analysis for the perturbed Chen–Lee–Liu equation, Opt. Quant. Electron., 55 (2023), 1151. https://doi.org/10.1007/s11082-023-05403-6 doi: 10.1007/s11082-023-05403-6
[31]	A. R. Seadawy, A. Ahmad, S. T. R. Rizvi, S. Ahmed, Bifurcation solitons, Y-type, distinct lumps and generalized breather in the thermophoretic motion equation via graphene sheets, Alex. Eng. J., 87 (2024), 374–388. https://doi.org/10.1016/j.aej.2023.12.023 doi: 10.1016/j.aej.2023.12.023
[32]	S. T. R. Rizvi, A. R. Seadawy, S. K. Naqvi, M. Ismail, Bifurcation analysis for mixed derivative nonlinear Schrödinger's equation, helix nonlinear Schrödinger's equation and Zoomeron model, Opt. Quant. Electron., 56 (2024), 452. https://doi.org/10.1007/s11082-023-06100-0 doi: 10.1007/s11082-023-06100-0
[33]	S. Kumar, H. Kharbanda, Chaotic behavior of predator-prey model with group defense and non-linear harvesting in prey, Chaos Soliton. Fract., 119 (2019), 19–28. https://doi.org/10.1016/j.chaos.2018.12.011 doi: 10.1016/j.chaos.2018.12.011
[34]	H. Kharbanda, S. Kumar, Chaos detection and optimal control in a cannibalistic prey-predator system with harvesting, Int. J. Bifurcat. Chaos, 30 (2020), 2050171. https://doi.org/10.1142/S0218127420501710 doi: 10.1142/S0218127420501710
[35]	S. Kumar, H. Kharbanda, Sensitivity and chaotic dynamics of an eco-epidemiological system with vaccination and migration in prey, Braz. J. Phys., 51 (2021), 986–1006. https://doi.org/10.1007/s13538-021-00862-2 doi: 10.1007/s13538-021-00862-2
[36]	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 Soliton. Fract., 171 (2023), 113436. https://doi.org/10.1016/j.chaos.2023.113436 doi: 10.1016/j.chaos.2023.113436
[37]	Z. Li, C. Huang, Bifurcation, phase portrait, chaotic pattern and optical soliton solutions of the conformable Fokas-Lenells model in optical fibers, Chaos Soliton. Fract., 169 (2023), 113237. https://doi.org/10.1016/j.chaos.2023.113237 doi: 10.1016/j.chaos.2023.113237
[38]	X. Zhang, F. H. Min, Y. P. Dou, Y. Y. Xu, Bifurcation analysis of a modified FitzHugh-Nagumo neuron with electric field, Chaos Soliton. Fract., 170 (2023), 113415. https://doi.org/10.1016/j.chaos.2023.113415 doi: 10.1016/j.chaos.2023.113415
[39]	A. Filiz, M. Ekici, A. Sonmezoglu, $F$-Expansion method and new exact solutions of the Schrödinger-KdV equation, Sci. World J., 2014 (2014), 534063. https://doi.org/10.1155/2014/534063 doi: 10.1155/2014/534063

Reader Comments

Your name:*

Email:*
© 2025 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)