Some New Traveling Wave Solutions of Nonlinear Fluid Models via the MSE Method

Abstract

In this study, some new exact wave solutions of nonlinear partial differential equations are investigated by the modified simple equation method. This method is applied to the $(2+1)$-dimensional Calogero-Bogoyavlenskii-Schiff equation and the $(3+1)$-dimensional Jimbo-Miwa equation. Our applications reveal how to use the proposed method to solve nonlinear partial differential equations with the balance number equal to two. Consequently, some new exact traveling wave solutions of these equations are achieved, and types of waves are determined. To verify our results and draw the graphs of the solutions, we use the Mathematica package program.

Keywords

• Calogero-Bogoyavlenskii- Schiff equation
• Jimbo-Miwa equation.
• Modified simple equation method
• Nonlinear partial differential equation
• Traveling wave solution

References

1. [1] L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Springer Science-Business Media, London, 2011.
2. [2] H. Jafari, N. Kadkhoda, Application of simplest equation method to the (2+1)-dimensional nonlinear evolution equations, New Trend Math. Sci., 2 (2014), 64-68.
3. [3] A. Tozar, A. Kurt, O. Tasbozan, New wave solutions of an integrable dispersive wave equation with a fractional time derivative arising in ocean engineering models, Kuwait J. Sci., 47 (2020), 22-33.
4. [4] A. Kurt, A. Tozar, O. Tasbozan, Applying the new extended direct algebraic method to solve the equation of obliquely interacting waves in shallow waters, J. Ocean Univ. China, 19 (2020), 772-780.
5. [5] A. Kurt, O. Tasbozan, H. Durur, The exact solutions of conformable fractional partial differential equations using new sub equation method, Fundam. J. Math. Appl., 2 (2019), 173-179.
6. [6] G. Bakıcıerler, S. Alfaqeih, E. Mısırlı, Analytic solutions of a (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation, Physica A, 582 (2021) Article ID 126255.
7. [7] E. M. E. Zayed, S. H. Ibrahim, Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method, Chin. Phys. Lett., 29 (2012), Article ID 060201.
8. [8] Y. S. Ozkan, E. Yasar, On the exact solutions of nonlinear evolution equations by the improved tan(j=2)-expansion method, Pramana, 94 (2020), 37.

9. [9] M. Cinar, I. Onder, A. Secer, A. Yusuf, T. A. Sulaiman, M. Bayram, H. Aydin, Soliton solutions of (2+1) dimensional Heisenberg ferromagnetic spin equation by the extended rational sine-cosine sine-cosine and sinh-cosh method, Int. J. Appl. Comput. Math., 7 (2021), 1-17.
10. [10] Y. Wen, Y. Xie, Exact solution of perturbed nonlinear Schr¨odinger equation using (G0=G;1=G)-expansion method, Pramana, 94 (2020), 18.
11. [11] M. S. Islam, M. A. Akbar, K. Khan, Analytical solutions of nonlinear Klein–Gordon equation using the improved F-expansion method, Opt. Quantum Electron., 50 (2018), 1-11.
12. [12] C. Cattani, T. A. Sulaiman, H. M. Baskonus, H. Bulut, Solitons in an inhomogeneous Murnaghan’s rod., Eur. Phys. J. Plus, 133 (2018), 228.
13. [13] S. Arshed, A. Biswas, A. K. Alzahrani, M. R. Belic, Solitons in nonlinear directional couplers with optical metamaterials by first integral method, Optik, 218 (2020), Article ID 165208.
14. [14] A. Ali, A. R. Seadawy, D. Lu, New solitary wave solutions of some nonlinear models and their applications, Adv. Differ. Equ., 1 (2018), 1-12.
15. [15] G. M. Moatimid, R. M. El-Shiekh, A. G. A. Al-Nowehy, Exact solutions for Calogero-Bogoyavlenskii-Schiff equation using symmetry method, Appl. Math. Comput., 220 (2013), 455-462.
16. [16] E. M. E. Zayed, Y. A. Amer, A. H. Arnous, Functional variable method and its applications for finding exact solutions of nonlinear PDEs in mathematical physics, Sci. Res. Essays., 8 (2013), 2068-2074.
17. [17] B. Ghanbari, K. S. Nisar, Determining new soliton solutions for a generalized nonlinear evolution equation using an effective analytical method, Alex. Eng. J., 59 (2020), 3171-3179.
18. [18] R. F. Zhang, M. C. Li, H. M. Yin, Rogue wave solutions and the bright and dark solitons of the (3+1)-dimensional Jimbo–Miwa equation, Nonlinear Dyn., 103 (2021), 1071-1079.
19. [19] M. S. Bruzon, M. L. Gandarias, C. Muriel, J. Ramirez, S. Saez, F. R. Romero, The Calogero-Bogoyavlenskii-Schiff equation in (2+1) dimensions, Theor. Math. Phys., 137 (2003), 1367–1377.
20. [20] M. H. Bashar, M. Roshid, Exact travelling wave solutions of the nonlinear evolution equations by improved F-expansion in mathematical physics, Commun. Math. Sci., 3 (2020), 115-123.
21. [21] H. M. Baskonus, T. A. Sulaiman, H. Bulut, New solitary wave solutions to the (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff and the Kadomtsev–Petviashvili hierarchy equations, Indian J. Phys., 91 (2017), 1237-1243.
22. [22] S. Kumar, D. Kumar, Lie symmetry analysis and dynamical structures of soliton solutions for the (2+1)-dimensional modified CBS equation, Int. J. Mod. Phys. B, 34 (2020), Article ID 2050221.
23. [23] S. M. Mabrouk, Traveling wave solutions of the extended Calogero-Bogoyavlenskii-Schiff equation, Int. J. Eng. Res. Technol., 8 (2019), 577-580.
24. [24] M. Usman, A. Nazir, T. Zubair, Z. Naheed, I. Rashid, S. T. Mohyud-Din, Solitary wave solutions of (2+1)-dimensional Davey-Stewartson equations by F-expansion method in terms of Weierstrass-Elliptic and Jacobian-Elliptic functions, Int. J. Mod. Math., 7 (2013), 149-169.
25. [25] H. D. Guo, T. C. Xia, B. B. Hu, High-order lumps, high-order breathers and hybrid solutions for an extended (3+1)-dimensional Jimbo–Miwa equation in fluid dynamics, Nonlinear Dyn., 100 (2020), 1-14.
26. [26] J. Liu, X. Yang, M. Cheng, Y. Feng, Y. Wang, Abound rogue wave type solutions to the extended (3+1)-dimensional Jimbo–Miwa equation, Comput. Math. Appl., 78 (2019), 1947-1959.
27. [27] F. H. Qi, Y. H. Huang, P. Wang, Solitary-wave and new exact solutions for an extended (3+1)-dimensional Jimbo-Miwa-like equation, Appl. Math. Lett., 100 (2020), Article ID 106004.
28. [28] X. Yin, L. Chen, J. Wang, X. Zhang, G. Ma, Investigation on breather waves and rogue waves in applied mechanics and physics, Alex. Eng. J., 60 (2021), 889-895.