Farjana Bilkis

^{1*}, Md. Emran Ali^{2}, Wahida Zaman Loskor^{1}, Samia Taher^{1}, Nusrat Jahan^{1}

Keywords:

Nonlinear evolution equation, Further extended tanh method, Generalized shallow water equation

Abstract: Further extended tanh (FET) method is suggested in this communication for solving generalized shallow water equation (GSWE). Based on the symbolic computational software, a realistic nonlinear integrable equation, GSWE that arises, typically, in atmospheric and ocean modelling, was reinvestigated to see its geometric feature using the FET method. The shock wave, soliton-like, kink type and periodic-like solution were found to be obtained. It is observed that the energy concentration of different wave profile is depended on the coefficient to Riccati equation. The obtained results were found to be somewhat similar with some of that obtained in the previous studies.

References

[1] Abdelsalam, U.M.: ‘Traveling wave solutions for shallow water equations’,Journal of Ocean Engineering and Science, 2017, 2, pp. 28-33

[2] Lü, Z., Zhang,H.: ‘On a further extended tanh method’, Physics Letters A, 2003, 307, (5-6), pp. 269-273

[3] Ablowitz, M. J., Clarkson, P. A.: ‘Solitons, nonlinear evolution equations and inverse scattering’, Cambridge university press, 1991, Vol. 149

[4] Hirota, R.: ‘Exact solution of the Korteweg—de Vries equation for multiple collisions of solitons’, Physical Review Letters, 1971, 27, (18), pp. 1192

[5] Weiss, J., Tabor, M., Carnevale, G.: ‘The Painlevé property for partial differential Equations’, Journal of Mathematical Physics, 1983, 24, (3), pp. 522-526

[6] Yan, C.: ‘A simple transformation for nonlinear waves’,Physics Letters A, 1996, 224, (1-2), pp. 77-84

[7] Wang, M.: ‘Exact solutions for a compound KdV-Burgers equation’, Physics Letters A,1996, 213, (5-6), pp. 279-287

[8] El-Shahed, M.: ‘Application of He’s homotopy perturbation method to Volterra’sintegro-differential equation’, International Journal of Nonlinear Sciences and Numerical Simulation, 2005, 6, (2), pp. 163-168

[9] He, J. H.: ‘Homotopy perturbation method for bifurcation of nonlinear problems’,International Journal of Nonlinear Sciences and Numerical Simulation, 2005a, 6, (2), pp. 207-208

[10] He, J. H.: ‘Application of homotopy perturbation method to nonlinear wave Equations’, Chaos, Solitons & Fractals, 2005b, 26, (3), pp. 695-700

[11] Abassy, T. A., El-Tawil, M. A., Saleh, H. K.: ‘The solution of KdV and mKdV equations using AdomianPade approximation’, International Journal of Nonlinear Sciences and Numerical Simulation, 2004, 5, (4), pp. 327-340

[12] Zayed, E. M. E., Zedan, H. A., Gepreel, K. A.: ‘Group analysis and modified extended tanh-function to find the invariant solutions and soliton solutions for nonlinear Euler equations’, International Journal of Nonlinear Sciences and Numerical Simulation, 2004, 5, (3), pp. 221-234

[13] Abdusalam, Η. A.: ‘On an improved complex tanh-function method’, International Journal of Nonlinear Sciences and Numerical Simulation, 2005, 6, (2), pp. 99-106

[14] Zhang, S., Tie-Cheng, X.: ‘Symbolic computation and new families of exact non-travelling wave solutions of (2+1)-dimensionalBroer–Kaup equations’, Communications in Theoretical Physics, 2006a, 45, (6), pp. 985

[15] Liu, J., Yang, K.: ‘The extended F-expansion method and exact solutions of nonlinear PDEs.’, Chaos, Solitons & Fractals, 2004, 22, (1), pp. 111-121

[16] Zhang, S., Xia, T.: ‘Symbolic computation and new families of exact non-travelling Wave solutions to (3+1)-dimensional Kadomstev–Petviashvili equation’, Applied mathematics and computation, 2006b, 181, (1), pp. 319-331

[17] Zhang, S.: ‘New exact solutions of the KdV–Burgers–Kuramoto equation’, Physics Letters A, 2006c, 358, (5-6), pp. 414-420

[18] Zhang, S.: ‘Symbolic computation and new families of exact non-travelling wave solutions of (2+1)-dimensional Konopelchenko–Dubrovsky equations’, Chaos, Solitons & Fractals, 2007a, 31, (4), pp. 951-959

[19] Zhang, S.: ‘The periodic wave solutions for the (2+1)-dimensional dispersive long water equation: Chaos, Solitons & Fractals’, 2007b, 32, (2), pp. 847-854

[20] He, J. H., Wu, X. H.: ‘Exp-function method for nonlinear wave equations’, Chaos, Solitons & Fractals, 2006, 30, (3), pp. 700-708

[21] He, J. H., Abdou, M. A.: ‘New periodic solutions for nonlinear evolution equations using Exp-function method’, Chaos, Solitons & Fractals, 2007, 34, (5), pp. 1421-1429

[22] Zhang, S., Xia, T.: ‘A generalized new auxiliary equation method and its applications to nonlinear partial differential equations’, Physics Letters A, 2007, 363, (5-6), pp. 356-360

[23] Wang, M., Li, X., Zhang, J.: ‘The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics’, Physics Letters A, 2008, 372, (4), pp. 417-423

[24] Zayed, E. M. E., Gepreel, K. A.: ‘The (G′/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics. Journal of Mathematical Physics’, 2009a, 50, (1), pp. 013502

[25] Zayed, E. M., Gepreel, K. A.: ‘Some applications of the G′/ G-expansion method to non-linear partial differential equations’, Applied Mathematics and Computation, 2009b, 212, (1), pp. 1-13

[26] Zayed, E. M. E., Gepreel, K. A.: ‘Three Types of Traveling Wave Solutions for Nonlinear Evolution Equations’, G. International Journal of Nonlinear Science, 2009c, 7, (4), pp. 501-512

[27] Zhang, S., Tong, J. L., Wang, W.: ‘A generalized (G/G)-expansion method for the mKdV equation with variable coefficients’, Physics Letters A, 2008a, 372, (13), pp. 2254-2257

[28] Zhang, J., Wei, X., Lu, Y.: ‘A generalized (G′/G)-expansion method and its applications’, Physics Letters A, 2008b, 372, (20), pp. 3653-3658

[29] Fan, E.: ‘Extended tanh-function method and its applications to nonlinear equations’, Physics Letters A, 2000, 277, (4-5), pp. 212-218

[30] Lü, Z. S., Zhang, H. Q.: ‘Soliton-like and period form solutions for highdimensional nonlinear evolution equations’, Chaos, Solitons & Fractals, 2003, 17, (4), pp. 669-673

[31] Malfliet, W.:‘Solitary wave solutions of nonlinear wave equation’, American Journal of Physics,1992,60, pp. 650

[32] Clarkson, P., Mansfield, E.: ‘On a shallow water wave equation, Nonlinearity’, 1994, 7, pp. 975-1000

[33] Tian, B., Gao, Y. T.: ‘Notiz: Report on the Generalized Tanh Method Extended to a Variable-Coefficient Korteweg-de Vries Equation’, ZeitschriftfürNaturforschung A, 1997, 52, (5), pp. 462-462

[34] Gao, Y. T., Tian, B.:‘Generalized tanh method with symbolic computation and generalized shallow water wave equation’ Computers & Mathematics with Applications, 1997, 33, (4), pp. 115-118

[35] Alquran, M., Jaradat, I., Sivasundaram, S., Al Shraiedeh, L.: ‘New shock-wave and periodic-wave solutions for some physical and engineering models’, Vakhnenko-Parkes, GEWB, GRLW and some integrable equations. Nonlinear Studies, 2020, 27, (2), pp. 393-403

[36] Tariqa, K. U., Seadawyd, A.: ‘Soliton solutions for (2+1) and (3+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony model equations and their applications’, Filomat, 2018, 32, (2), pp. 531–542