Dynamics of Soliton Solutions and Various Evolving Formations of the Jaulent–Miodek and Zakharov–Kuzetsov Equations Utilizing the Newly Proposed Extended Generalized Approach
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Setu Rani [1] ; Sachin Kumar [1]
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[1] University of Delhi
University of Delhi
India
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- Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 2, 2025
- Idioma: inglés
- Enlaces
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Resumen
This research introduces a new approach, the extended Generalized Exponential Rational Differential Function (eGERDF) technique, for finding the closed-form analytical solutions of higher-order nonlinear partial differential equations such as the integrodifferential Jaulent–Miodek equation and the Zakharov–Kuzetsov equation. Building upon the generalized exponential rational function method, we developed an advanced trial solution incorporating extra terms, including functions and their derivatives. This enhancement makes the new method more generalized and practical for solving highly nonlinear partial differential equations. The obtained closed-form solutions includes rational, hyperbolic, and trigonometric functions, as well as a few free parameters, which are critical for illustrating and exhibiting the different dynamics of soliton solutions of the Jaulent–Miodek and Zakharov–Kuzetsov models. To the best of our knowledge, these solutions have been obtained for the first time in the literature using this extended technique. The results include graphical representations of the solutions for both equations, displayed in various formats such as 3D, 2D, and contour plots. These visualizations showcase multiple waveforms, including periodic waves, multi-peakons, single-soliton, lumps, rough waves, and multi-solitons, along with their interactions using numerical simulation. Such solutions play a critical role in advancing knowledge across multiple fields, including condensed matter physics, materials science, plasma physics, fluid dynamics, environmental modeling, and diverse engineering systems. The obtained findings demonstrate that the proposed method is more useful, valid, and effective, indicating its practical application.
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Referencias bibliográficas
- 1. Wang, K.J., Wang, G.D., Shi, F., Liu, X.L., Zhu, H.W.: Variational principle, Hamiltonian, bifurcation analysis, chaotic behaviors and...
- 2. Wang, K.J., Li, S.: Novel complexiton solutions to the new extended (3+ 1)-dimensional Boiti-LeonManna-Pempinelli equation for incompressible...
- 3. Wang, K.J.: The generalized (3+ 1)-dimensional B-type Kadomtsev-Petviashvili equation: resonant multiple soliton, N-soliton, soliton...
- 4. Wang, K.J.: N-soliton, soliton molecules, Y-type soliton, periodic lump and other wave solutions of the new reduced B-type Kadomtsev-Petviashvili...
- 5. Elboree, M.K.: The Jacobi elliptic function method and its application for two component BKP hierarchy equations. Comput. Math. Appl. 62,...
- 6. Tarla, S., Ali, K.K., Yilmazer, R., et al.: New optical solitons based on the perturbed Chen-Lee-Liu model through Jacobi elliptic function...
- 7. Kumar, S., Rani, S.: Invariance analysis, optimal system, closed-form solutions and dynamical wave structures of a (2+1)-dimensional...
- 8. Hosseini, K., Hınçal, E., Alizadeh, F., et al.: Bifurcation analysis, sensitivity analysis, and Jacobi elliptic function structures to...
- 9. Seadawy, A.R., Ali, A., Bekir, A.: Novel solitary waves solutions of the extended cubic (3+1)- dimensional Schrödinger equation via...
- 10. Osman, M.S.: On complex wave solutions governed by the 2D Ginzburg-Landau equation with variable coefficients. Optik 156, 169–174 (2018)
- 11. Baskonus, H.M., Younis, M., Bilal, M., Younas, U., Shafqat-ur-Rehman, G.: Modulation instability analysis and perturbed optical soliton...
- 12. Wazwaz, A.M., Albalawi, W., Tantawy, S.: Optical envelope soliton solutions for coupled nonlinear Schrödinger equations applicable to...
- 13. Tasnim, F., Akbar, A., Osman, M.S.: The extended direct algebraic method for extracting analytical solitons solutions to the cubic nonlinear...
- 14. Ghanbari, B., Inc, M.: A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear...
- 15. Younas, U., Seadawy, A.R., Younis, M., Rizvi, S.: Construction of analytical wave solutions to the conformable fractional dynamical system...
- 16. Ullah, N., Asjad, M.I., Almusawa, M.Y., Eldin, S.M.: Dynamics of nonlinear optics with different analytical approaches. Fractal Fract....
- 17. Rehman, H.U., Akber, R., Wazwaz, A.M., Alshehri, H.M., Osman, M.S.: Analysis of Brownian motion in stochastic Schrödinger wave equation...
- 18. Younas, U., Sulaiman, T.A., Ren, J.: On the optical soliton structures in the magneto electro-elastic circular rod modeled by nonlinear...
- 19. Raza, N., Osman, M.S., Abdel-Aty, A.H., et al.: Optical solitons of space-time fractional Fokas-Lenells equation with two versatile integration...
- 20. Belousov, N.: Bäcklund transformation for the nonlinear Schrödinger equation. J. Math. Sci. 264, 203–214 (2022)
- 21. Wang, K.J., Shi, F., Li, S., Li, G., Xu, P.: Resonant Y-type soliton, interaction wave and other wave solutions to the (3+ 1)-dimensional...
- 22. Younas, U., Sulaiman, T.A., Ren, J.: On the collision phenomena to the (3+1)-dimensional generalized nonlinear evolution equation:...
- 23. Younas, U., Sulaiman, T.A., Ren, J., Yusuf, A.: Lump interaction phenomena to the nonlinear ill-posed Boussinesq dynamical wave equation....
- 24. Dhiman, S.K., Kumar, S.: Analyzing specific waves and various dynamics of multi-peakons in (3+1)- dimensional p-type equation using...
- 25. Cakicioglu, H., Ozisik, M., Secer, A., Bayram, M.: Kink soliton dynamic of the (2+1)-dimensional integro-differential Jaulent-Miodek...
- 26. Jaulent, M., Miodek, I.: Nonlinear evolution equations associated with ‘enegry-dependent Schrödinger potentials’. Lett. Math. Phys. 1,...
- 27. Wazwaz, A.-M.: The tanh − coth and the sech methods for exact solutions of the Jaulent-Miodek equation. Phys. Lett. A 366, 85–90 (2007)
- 28. Ali, K.K., Tarla, S.: Bifurcations of phase portraits and exact solutions of the (2+1)-dimensional integro-differential Jaulent-Miodek...
- 29. Kaewta, S., Sirisubtawee, S., Khansai, N.: Explicit exact solutions of the (2+1)-dimensional integrodifferential Jaulent-Miodek evolution...
- 30. He, J.-H., Zhang, L.-N.: Generalized solitary solution and compacton-like solution of the JaulentMiodek equations using the Exp-function...
- 31. Wazwaz, A.M.: The extended tanh method for the Zakharov-Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms....
- 32. Li, B., Chen, Y., Zhang, H.: Exact travelling wave solutions for a generalized Zakharov-Kuznetsov equation. Appl. Math. Comput. 146, 653–666...
- 33. Iqbal, M., Seadawy, A.R., Lu, D., Xia, X.: Construction of bright-dark solitons and ion-acoustic solitary wave solutions of dynamical...
- 34. Alammari, M., Iqbal, M., Ibrahim, S., et al.: Exploration of solitary waves and periodic optical soliton solutions to the nonlinear two...
- 35. Durur, H., Yoku¸s, A., Duran, S.: Investigation of exact soliton solutions of nematicons in liquid crystals according to nonlinearity...
- 36. Kumar, S., Niwas, M.: Analyzing multi-peak and lump solutions of the variable-coefficient Boiti-LeonManna-Pempinelli equation: a comparative...
- 37. Adeyemo, O.D.: Self-Adjointness, dynamics of solutions and conserved currents of a multidimensional nonlinear fifth-order generalized...
