Basitleştirilmiş Modifiye Camassa–Holm Denkleminin Soliton Çözümleri Üzerine Yeni Bir Çalışma

Abstract

In this study, we investigate the soliton solutions of the simplified modified Camassa–Holm equation, which plays a significant role in various applications within mathematical physics and engineering disciplines. By applying an appropriate wave transformation, the equation is reduced to an ordinary differential equation form. To obtain its analytical solutions, we employ the generalized Kudryashov method — a technique widely adopted by researchers for its strong ability to generate rational solutions and encompass a broad variety of functional forms. The application of this method produces diverse types of soliton solutions, including bright, dark, kink, and singular profiles. These solutions are verified through direct substitution and illustrated graphically, revealing distinct behaviors under varying parameter conditions. The results uncover new wave structures not previously reported for the simplified modified Camassa–Holm equation, offering deeper insight into its nonlinear dynamics. The novelty of this study lies in the first successful application of the generalized Kudryashov method to this model, highlighting its efficiency, practicality, and wide applicability to nonlinear differential equations.

Keywords

• Soliton çözümleri
• Genelleştirilmiş Kudryashov yöntemi
• basitleştirilmiş modifiye Camassa–Holm denklemi
• dalga dönüşümü
• kink soliton

A Novel Study of The Soliton Solutions of the Simplified Modified Camassa-Holm Equation

Abstract

In this study, we investigate the soliton solutions of the simplified modified Camassa–Holm equation, which plays a significant role in various applications within mathematical physics and engineering disciplines. By applying an appropriate wave transformation, the equation is reduced to an ordinary differential equation form. To obtain its analytical solutions, we employ the generalized Kudryashov method — a technique widely adopted by researchers for its strong ability to generate rational solutions and encompass a broad variety of functional forms. The application of this method produces diverse types of soliton solutions, including bright, dark, kink, and singular profiles. These solutions are verified through direct substitution and illustrated graphically, revealing distinct behaviors under varying parameter conditions. The results uncover new wave structures not previously reported for the simplified modified Camassa–Holm equation, offering deeper insight into its nonlinear dynamics. The novelty of this study lies in the first successful application of the generalized Kudryashov method to this model, highlighting its efficiency, practicality, and wide applicability to nonlinear differential equations.

Keywords

• Soliton solutions
• the generalized Kudryashov method
• the simplified modified Camassa-Holm equation
• wave transform
• kink soliton.

References

1. [1] Triki, H., Babatin, M.M., Biswas, A., “Chirped bright solitons for Chen–Lee–Liu equation in optical fibers and PCF”, Optik 149 (2017) : 300–303.
2. [2] Karakoç, S.B.G., Ali, K.K., “Theoretical and computational structures on solitary wave solutions of Benjamin Bona Mahony-Burgers equation”, Tbilisi Math. J. 14(2) (2021) : 33–50.
3. [3] Zhao, Z., He, L., “Resonance Y-type soliton and hybrid solutions of a (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation”, Appl. Math. Lett. 122 (2021) : 107497.
4. [4] Nasreen, N., Yadav, A., Malik, S., Hussain, E., Alsubaie, A.S., Alsharif, F., “Phase trajectories, chaotic behavior, and solitary wave solutions for (3+1)-dimensional integrable Kadomtsev–Petviashvili equation in fluid dynamics”, Chaos Solitons Fractals 188 (2024) : 115588.
5. [5] Başhan, A., “A novel outlook to the mKdV equation using the advantages of a mixed method”, Applicable Analysis 102(1) (2021) : 65–87.
6. [6] Cevikel, A.C., Bekir, A., Guner, O., “Exploration of new solitons solutions for the Fitzhugh-Nagumo-type equations with conformable derivatives”, Int. J. Mod. Phys. B 37(23) (2023) : 2350224.
7. [7] Mahmood, A., Srivastava, H.M., Abbas, M., Abdullah, F.A., Mohammed, P.O., Baleanu, D., Chorfi, N., “Optical soliton solutions of the coupled Radhakrishnan-Kundu-Lakshmanan equation by using the extended direct algebraic approach”, Heliyon 9(10) (2023) : e20852.
8. [8] Biswas, A., Hubert, M.B., Justin, M., Betchewe, G., Doka, S.Y., Crepin, K.T., Ekici, M., Belic, M., “Chirped dispersive bright and singular optical solitons with Schrödinger–Hirota equation”, Optik 168 (2018) : 192–195.

9. [9] Kudryashov, N.A., “Painlevé analysis of the Sasa–Satsuma equation”, Phys. Lett. A 525, (2024) : 129900.
10. [10] Kilic, B., Inc, M., “The First Integral Method for the time fractional Kaup-Boussinesq System with time dependent coefficient”, Appl. Math. Comput. 254 (2015) : 70–74.
11. [11] Saha, A., Karakoç, S.B.G., Ali, K.K., “New exact soliton solutions bifurcation and multistability behaviors of traveling waves for the (3+1)-dimensional modified Zakharov-Kuznetsov equation with higher order dispersion”, Math. Sci. and App. E-Notes 11(4) (2023) : 226–240.
12. [12] Zhou, Q., Ekici, M., Sonmezoglu, A., Mirzazadeh, M., “Optical solitons with Biswas-Milovic equation by extended G'/G-expansion method”, Optik 127 (2016) : 6277–6290.
13. [13] Başhan, A. "Bell-shaped soliton solutions and travelling wave solutions of the fifth-order nonlinear modified Kawahara equation", Int. J. Nonlinear Sci. Numer. Simul. 22(6) (2021) : 781-795.
14. [14] Karakoç, S.B.G., Ali, K.K., Sucu, D., “A new perspective for analytical and numerical soliton solutions of the Kaup Kupershmidt and Ito equations”, J. of Comp. and Appl. Math. 421 (2023) : 114850.
15. [15] Arshed, S., Biswas, A., Alzahrani, A.K., Belic, M.R., “Solitons in nonlinear directional couplers with optical metamaterials by first integral method”, Optik 218 (2020) : 165208.
16. [16] Ali, K.K., Karakoç, S.B.G., Rezazadeh, H., “Optical Soliton Solutions of the Fractional Perturbed Nonlinear Schrodinger Equation”, TWMS J. of Appl. and Eng. Math. 10(4) (2020) : 930–939.
17. [17] Mathanaranjan, T., “Solitary wave solutions of the Camassa–Holm-Nonlinear Schrödinger Equation”, Results Phys. 19 (2020) : 103549.
18. [18] Yasin, S., Khan, A., Ahmad, S., Osman, M.S., “New exact solutions of (3+1)-dimensional modified KdV-Zakharov-Kuznetsov equation by Sardar-subequation method”, Opt. Quantum Electron. 56(1) (2024) : 90.
19. [19] Ma, Y.L., Wazwaz, A.M., Li, B.Q., “Novel bifurcation solitons for an extended Kadomtsev–Petviashvili equation in fluids”, Phys. Lett. A 413 (2021) : 127585.
20. [20] Mathanaranjan, T., Rezazadeh, H., Şenol, M., Akinyemi, L., “Optical singular and dark solitons to the nonlinear Schrödinger equation in magneto-optic waveguides with anti-cubic nonlinearity” Opt. Quantum Electron. 53(12) (2021) : 722.
21. [21] Li, W.W., Tian, Y., Zhang, Z., “F-expansion method and its application for finding new exact solutions to the sine-Gordon and sinh-Gordon equations”, Appl. Math. Comput. 219(3) (2012) : 1135–1143.
22. [22] Almatrafi, M.B., “Solitary wave solutions to a fractional model using the improved modified extended tanh-function method”, Fractal Fract. 7(3) (2023) : 252.
23. [23] Başhan, A., "Highly efficient approach to numerical solutions of two different forms of the modified Kawahara equation via contribution of two effective methods", Math. and Comp. in Simulation 179 (2021) : 111-125.
24. [24] Gupta, S., Singh, J., Kumar, D., “Application of homotopy perturbation transform method for solving time-dependent functional differential equations”, Int. J. Nonlin. Sci. 16 (2013) : 37–49.
25. [25] Wang, K.L., “New mathematical approaches to nonlinear coupled Davey–Stewartson Fokas system arising in optical fibers”, Math. Methods Appl. Sci. 47 (2024) : 12668–12683.
26. [26] Başhan, A., "Modification of quintic B-spline differential quadrature method to nonlinear Korteweg-de Vries equation and numerical experiments", Appl. Numerical Math. 167 (2021) : 356-374.
27. [27] Camassa, R., Holm, D.D., “An integrable shallow water equation with peaked solitons”, Phys. Rev. Lett. 71 (1993) : 1661–1664.
28. [28] Tian, L., Song, X., “New peaked solitary wave solutions of the generalized Camassa-Holm equation”, Chaos Solitons Fractals 19(3) (2004) : 621–637.
29. [29] Feng, B.F., Hu, H.C., Sheng, H.H., Yin, W., Yu, G.F., “Integrable semi-discretization for a modified Camassa-Holm equation with cubic nonlinearity”, Symmetry Integrability Geom. Methods Appl. (SIGMA) 20 (2024) : 091.
30. [30] Aziz, M.A., Iqbal, M.A., Akbar, M.A., “Exploring soliton and soliton-type solutions to the modified Camassa-Holm and Schrödinger-Hirota equations: an analytical approach”, Phys. Scr. 100(2) (2025) : 025234.
31. [31] Zafar, A., Raheel, M., Hosseini, K., Mirzazadeh, M., Salahshour, S., Park, C., Shin, D.Y., “Diverse approaches to search for solitary wave solutions of the fractional modified Camassa–Holm equation”, Results Phys. 31 (2021) : 104882.
32. [32] Xi, X., Hu, W., Tang, B., Deng, P., Qiao, Z., “Multi-symplectic method for the two-component Camassa–Holm (2CH) system”, J. Nonlinear Math. Phys. 31(1) (2024) : 50.
33. [33] Wazwaz, A.M., “New compact and noncompact solutions for two variants of a modified Camassa-Holm equation”, Appl. Math. Comput. 163(3) (2005) : 1165–1179.
34. [34] Irshad, A., Usman, M., Mohyud-Din, S.T., “Exp-function method for simplified modified Camassa-Holm equation”, Int. J. Mod. Math. Sci. 4(2) (2012) : 146–155.
35. [35] Javeed, S., Abbasi, M.A., Imran, T., Fayyaz, R., Ahmad, H., Botmart, T., “New soliton solutions of simplified modified Camassa Holm equation, Klein–Gordon–Zakharov equation using First Integral Method and Exponential Function Method”, Results Phys. 38 (2022) : 105506.
36. [36] Liu, X., “The stability of exact solitary wave solutions for simplified modified Camassa–Holm equation”, Commun. Nonlinear Sci. Numer. Simul. 108 (2022) : 106224.
37. [37] Islam, S.M.R., Arafat, S.M.Y., Wang, H., “Abundant closed-form wave solutions to the simplified modified Camassa-Holm equation”, J. Ocean Eng. Sci. 8 (2023) : 238–245.
38. [38] Devnath, S., Khan, S., Akbar, M.A., “Exploring solitary wave solutions to the simplified modified Camassa-Holm equation through a couple sophisticated analytical approaches”, Results Phys. 59 (2024) : 107580.
39. [39] Sadiq, S., Javid, A., “Novel solitary wave solutions in dual-mode simplified modified Camassa-Holm equation in shallow water waves”, Opt. Qua. Electron. 56(3) (2024) : 464.
40. [40] Zaman, U.H.M., Arefin, M.A., Akbar, M.A., Uddin, M.H., “Diverse soliton wave profile analysis in ion-acoustic wave through an improved analytical approach”, Partial Differ. Equ. Appl. Math. 12 (2024 ) : 100932.
41. [41] Koç, D., Kılbitmez, S., Bulut, H., “The new complex travelling wave solutions of the simplified modified Camassa-Holm equation”, Opt. Quantum Electron. 56 (2024) : 215.