Application of the rational \((G^\prime/G)\)-expansion method for solving some coupled and combined wave equations. (English) Zbl 1491.35111
Summary: In this paper, we explore the travelling wave solutions for some nonlinear partial differential equations by using the recently established rational \((G^\prime/G)\)-expansion method. We apply this method to the combined KdV-mKdV equation, the reaction-diffusion equation and the coupled Hirota-Satsuma KdV equations. The travelling wave solutions are expressed by hyperbolic functions, trigonometric functions and rational functions. When the parameters are taken as special values, the solitary waves are also derived from the travelling waves. We have also given some figures for the solutions.
MSC:
| 35C07 | Traveling wave solutions |
| 35C08 | Soliton solutions |
| 35A22 | Transform methods (e.g., integral transforms) applied to PDEs |
| 35L71 | Second-order semilinear hyperbolic equations |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
References:
| [1] | Ablowitz, M. J., Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (Vol. 149), Cambridge University Press, 1991. https://doi.org/10.1017/cbo9780511623998 · Zbl 0762.35001 |
| [2] | Hirota, R., Exact n-soliton solutions of the wave equation of long waves in shallowwater and in nonlinear lattices, Journal of Mathematical Physics, 14(7) (1973), 810-814. https://doi.org/10.1063/1.1666400 · Zbl 0261.76008 |
| [3] | Cariello, F., Tabor, M., Similarity reductions from extended Painlev´e expansions for nonintegrable evolution equations, Physica D: Nonlinear Phenomena, 53(1) (1991), 59-70. https://doi.org/10.1016/0167-2789(91)90164-5 · Zbl 0745.35046 |
| [4] | Fan, E., Extended tanh-function method and its applications to nonlinear equations, Physics Letters A, 227(4) (2000), 212-218. https://doi.org/10.1016/S0375-9601(00)00725-8 · Zbl 1167.35331 |
| [5] | Liu, S., Fu, Z., Liu, S., Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Physics Letters A, 289(1) (2001), 69-74. https://doi.org/10.1016/S0375-9601(01)00580-1 · Zbl 0972.35062 |
| [6] | Wang, M., Solitary wave solutions for variant Boussinesq equations, Physics Letters A, 199(3-4) (1995), 169-172. https://doi.org/10.1016/0375-9601(95)00092-H · Zbl 1020.35528 |
| [7] | Wang, M., Exact solutions for a compound KdV-Burgers equation, Physics Letters A, 213(5-6) (1996), 279-287. https://doi.org/10.1016/0375-9601(96)00103-X · Zbl 0972.35526 |
| [8] | Wang, M., Zhou, Y., Li, Z., Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A, 216(1-5) (1996), 67-75. https://doi.org/10.1016/0375-9601(96)00283-6 · Zbl 1125.35401 |
| [9] | Kudryashov, N. A., Exact solutions of the generalized Kuramoto-Sivashinsky equation, Physics Letters A, 147(5-6) (1990), 287-291. https://doi.org/10.1016/0375-9601(90)90449X. |
| [10] | He, J. H., Wu, X. H., Exp-function method for nonlinear wave equations, Chaos, Solitons Fractals, 30(3) (2006), 700-708. https://doi.org/10.1016/j.chaos.2006.03.020 · Zbl 1141.35448 |
| [11] | Rogers, C., Shadwick, W. F., B¨acklund Transformations and Their Applications. Academic Press, New York, USA, 1982. · Zbl 0492.58002 |
| [12] | Yang, L., Liu, J., Yang, K., Exact solutions of nonlinear PDE, nonlinear transformations and reduction of nonlinear PDE to a quadrature, Physics Letters A, 278(5) (2001), 267-270. https://doi.org/10.1016/S0375-9601(00)00778-7 · Zbl 0972.35003 |
| [13] | Yan, Z., Zhang, H., New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water, Physics Letters A, 285(5) (2001), 355-362. https://doi.org/10.1016/S0375-9601(01)00376-0 · Zbl 0969.76518 |
| [14] | Yan, Z., New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations, Physics Letters A, 292(1) (2001), 100-106. https://doi.org/10.1016/S0375-9601(01)00772-1 · Zbl 1092.35524 |
| [15] | 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, 372(4) (2008), 417-423. https://doi.org/10.1016/j.physleta.2007.07.051 · Zbl 1217.76023 |
| [16] | Zhang, S., Tong, J. L., Wang, W., A generalized (G’/G)-expansion method for the mKdV equation with variable coefficients, Physics Letters A, 372(13) (2008), 2254-2257. https://doi.org/10.1016/j.physleta.2007.11.026 · Zbl 1220.37072 |
| [17] | Zhang, J. L., Wang, M. L., Li, X. Z., The subsidiary ordinary differential equations and the exact solutions of the higher order dispersive nonlinear Schrodinger equation, Physics Letters A, 357(3) (2006), 188-195. https://doi.org/10.1016/j.physleta.2006.03.081 · Zbl 1236.81092 |
| [18] | Wang, M., Li, X., Zhang, J., Various exact solutions of nonlinear Schrodinger equation with two nonlinear terms, Chaos, Solitons Fractals, 31(3) (2007), 594-601. https://doi.org/10.1016/j.chaos.2005.10.009 · Zbl 1138.35411 |
| [19] | Li, X., Wang, M., A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with high-order nonlinear terms, Physics Letters A, 361(1) (2007), 115-118. https://doi.org/10.1016/j.physleta.2006.09.022 · Zbl 1170.35085 |
| [20] | Wang, M., Li, X., Zhang, J., Sub-ODE method and solitary wave solutions for higher order nonlinear Schr¨odinger equation, Physics Letters A, 363(1) (2007), 96-101. https://doi.org/10.1016/j.physleta.2006.10.077 · Zbl 1197.81129 |
| [21] | Islam, M., Akbar, M. A., Azad, A. K., A Rational (G’/G)-expansion method and its application to the modified KdV-Burgers equation and the (2+ l)-dimensional Boussinesq equation, Nonlinear Studies, 22(4) (2015), 635-645. · Zbl 1335.35018 |
| [22] | Konno, K., Ichikawa Y. H., A modified Korteweg de Vries equation for ion acoustic waves, Journal of the Physical Society of Japan, 37(6) (1974), 1631-1636. https://doi.org/10.1143/JPSJ.37.1631 |
| [23] | Narayanamurti, V., Varma, C. M., Nonlinear propagation of heat pulses in solids, Physical Review Letters, 25(16) (1970), 1105. https://doi.org/10.1103/PhysRevLett.25.1105 |
| [24] | Tappert, F. D., Varma, C. M., Asymptotic theory of self-trapping of heat pulses in solids, Physical Review Letters, 25(16) (1970), 1108. https://doi.org/10.1103/PhysRevLett.25.1108 |
| [25] | Yomba, E., The extended Fan’s sub-equation method and its application to KdV-MKdV, BKK and variant Boussinesq equations, Physics Letters A, 336(6) (2005), 463-476. https://doi.org/10.1016/j.physleta.2005.01.027 · Zbl 1136.35451 |
| [26] | Fan, E., Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos, Solitons Fractals, 16(5) (2003), 819-839. https://doi.org/10.1016/S0960-0779(02)00472-1 · Zbl 1030.35136 |
| [27] | Wadati, M., Wave propagation in nonlinear lattice, II. Journal of the Physical Society of Japan, 38(3) (1975), 681-686. https://doi.org/10.1143/JPSJ.38.673 · Zbl 1334.82023 |
| [28] | Mohamad, M. N. B., Exact solutions to the combined KdV and MKdV equation, Mathematical Methods in the Applied Sciences, 15(2) (1992), 73-78. https://doi.org/10.1002/mma.1670150202 · Zbl 0741.35071 |
| [29] | 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, 50(1) (2009), 013502. https://doi.org/10.1063/1.3033750 |
| [30] | Mei, J. Q., Zhang, H. Q., Jiang, D. M., New exact solutions for a reaction-diffusion equation and a Quasi-Camassa Holm equation, Appl. Math. E-Notes, 4 (2004), 85-91. · Zbl 1064.35032 |
| [31] | Wu, Y., Geng, X., Hu, X., Zhu, S., A generalized Hirota-Satsuma coupled Korteweg-de Vries equation and Miura transformations, BKK and variant Boussinesq equations, Physics Letters A, 255(4-6) (1999), 259-264. https://doi.org/10.1016/S0375-9601(99)00163-2 · Zbl 0935.37029 |
| [32] | Inan, I. E., Duran, S., Ugurlu, Y., \(Tan(F(\frac{\xi }{2}))\)-expansion method for traveling wave solutions of AKNS and Burgers-like equations, Optik, 138 (2017), 15-20. https://doi.org/10.1016/j.ijleo.2017.02.087 |
| [33] | Ekici, M., Ayaz, F., Solution of model equation of completely passive natural convection by improved differential transform method, Research on Engineering Structures and Materials, 3(1) (2017), 1-10. http://dx.doi.org/10.17515/resm2015.10me0818 |
| [34] | Ekici, M., Ünal, M., Application of the Exponential Rational Function Method to Some Fractional Soliton Equations, In Emerging Applications of Differential Equations and Game Theory, (pp. 13-32), IGI Global, 2020. |
| [35] | Ünal, M., Ekici, M., The double (G’/G, 1/G)-expansion method and its applications for some nonlinear partial differential equations, Journal of the Institute of Science and Technology, 11(1) (2021), 599-608. https://doi.org/10.21597/jist.767930 |
| [36] | Islam, M. T., Akter, M. A., Distinct solutions of nonlinear space-time fractional evolution equations appearing in mathematical physics via a new technique, Partial Differential Equations in Applied Mathematics, 3 (2021), 100031. https://doi.org/10.1016/j.padiff.2021.100031 · Zbl 1488.34040 |
| [37] | Islam, M. T., Akter, M. A., Exact analytic wave solutions to some nonlinear fractional differential equations for the shallow water wave arise in physics and engineering, Journal of Research in Engineering and Applied Sciences, 6(1) (2021), 11-18. |
| [38] | Islam, T., Akter, A., Further fresh and general traveling wave solutions to some fractional order nonlinear evolution equations in mathematical physics, Arab Journal of Mathematical Sciences, 26(1/2) (2020), Doi: 10.1108/AJMS-09.2020-0078 · Zbl 1488.34040 |
| [39] | Akbar, M. A., Ali, N. H. M., Islam, M. T., Multiple closed form solutions to some fractional order nonlinear evolution equations in physics and plasma physics, AIMS Mathematics, 4(3) (2019), 397-411. doi: 10.3934/math.2019.3.397 · Zbl 1484.35370 |
| [40] | Islam, M. T., Akbar, M. A., Azad, M. A. K., Closed-form travelling wave solutions to the nonlinear space-time fractional coupled Burgers’ equation, Arab Journal of Basic and Applied Sciences, 26(1) (2019), 1-11. https://doi.org/10.1080/25765299.2018.1523702 |
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