Extended homogeneous balance conditions in the sub-equation method. (English) Zbl 1491.35102
Summary: The sub-equation method is a kind of straightforward algebraic method to construct exact solutions of nonlinear evolution equations. In this paper, the sub-equation method is improved by proposing some extended homogeneous balance conditions. By applying them to several examples, it can be seen that new solutions could indeed be obtained.
Keywords:
sub-equation method; homogeneous balance conditions; Jacobi elliptic function; Riccati equationSoftware:
RATHReferences:
| [1] | M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc. Lecture Note Ser. 149, Cambridge University, Cambridge, 1991. · Zbl 0762.35001 |
| [2] | M. M. A. El-Sheikh, H. M. Ahmed, A. H. Arnous and W. B. Rabie, Optical solitons and other solutions in birefringent fibers with Biswas-Arshed equation by Jacobi’s elliptic function approach, Optik 202 (2020), Article ID 163546. |
| [3] | E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000), no. 4-5, 212-218. · Zbl 1167.35331 |
| [4] | K. A. Gepreel, T. A. Nofal and A. A. Al-Asmari, Abundant travelling wave solutions for nonlinear Kawahara partial differential equation using extended trial equation method, Int. J. Comput. Math. 96 (2019), no. 7, 1357-1376. · Zbl 1499.35535 |
| [5] | K. Hosseini, M. Inc, M. Shafiee, M. Ilie, A. Shafaroody, A. Yusuf and M. Bayram, Invariant subspaces, exact solutions and stability analysis of nonlinear water wave equations, J. Ocean Eng. Sci. 5 (2020), 35-40. |
| [6] | X. B. Hu and H. W. Tam, New integrable differential-difference systems: Lax pairs, bilinear forms and soliton solutions, Inverse Problems 17 (2001), no. 2, 319-327. · Zbl 0985.37086 |
| [7] | I. A. Kunin, Elastic Media with Microstructure. I: One-Dimensional Models, Springer Ser. Solid-State Sci. 26, Springer, Berlin, 1982. · Zbl 0527.73002 |
| [8] | Z. B. Li and Y. P. Liu, RATH: A Maple package for finding travelling solitary wave solutions to nonlinear evolution equations, Comput. Phys. Comm. 148 (2002), no. 2, 256-266. · Zbl 1196.35008 |
| [9] | X. Liu and C. Liu, The relationship among the solutions of two auxiliary ordinary differential equations, Chaos Solitons Fractals 39 (2009), no. 4, 1915-1919. · Zbl 1197.34021 |
| [10] | Z. Y. Long, L. Y. Ping and L. Z. Bin, A connection between the (G^{\prime}/G)-expansion method and the truncated Painlevé expansion method and its application to the mKdV equation, Chinese Phys. B 19 (2010), no. 3, Article ID 030306. |
| [11] | W. X. Ma, Y. Zhang, Y. Tang and J. Tu, Hirota bilinear equations with linear subspaces of solutions, Appl. Math. Comput. 218 (2012), no. 13, 7174-7183. · Zbl 1245.35109 |
| [12] | W. Malfliet, Solitary wave solutions of nonlinear wave equations, Amer. J. Phys. 60 (1992), no. 7, 650-654. · Zbl 1219.35246 |
| [13] | A. P. Márquez and M. S. Bruzón, Travelling wave solutions of a one-dimensional viscoelasticity model, Int. J. Comput. Math. 97 (2020), no. 1-2, 30-39. · Zbl 1479.35856 |
| [14] | V. B. Matveev and V. B. Matveev, Darboux Transformations and Solitons, Springer, Berlin, 1991. · Zbl 0744.35045 |
| [15] | A. V. Mikhailov, The reduction problem and the inverse scattering method, Phys. D 3 (1981), no. 1-2, 73-117. · Zbl 1194.37113 |
| [16] | R. C. Mittal and S. Pandit, Sensitivity analysis of shock wave Burgers’ equation via a novel algorithm based on scale-3 Haar wavelets, Int. J. Comput. Math. 95 (2018), no. 3, 601-625. · Zbl 1390.65162 |
| [17] | Y. Z. Peng, Exact solutions for some nonlinear partial differential equations, Phys. Lett. A 314 (2003), no. 5-6, 401-408. · Zbl 1040.35102 |
| [18] | S. Sahoo and S. Saha Ray, Solitary wave solutions for time fractional third order modified KdV equation using two reliable techniques (G^{\prime}/G)-expansion method and improved (G^{\prime}/G)-expansion method, Phys. A 448 (2016), 265-282. · Zbl 1400.35204 |
| [19] | M. Wadati, K. Konno and Y. H. Ichikawa, A generalization of inverse scattering method, J. Phys. Soc. Japan 46 (1979), no. 6, 1965-1966. · Zbl 1334.81106 |
| [20] | A. M. Wazwaz, The Hirota’s bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equation, Appl. Math. Comput. 200 (2008), no. 1, 160-166. · Zbl 1153.65364 |
| [21] | G. Q. Xu, New types of exact solutions for the fourth-order dispersive cubic-quintic nonlinear Schrödinger equation, Appl. Math. Comput. 217 (2011), no. 12, 5967-5971. · Zbl 1210.35241 |
| [22] | R. X. Yao, W. Wang and T. H. Chen, New solutions of three nonlinear space- and time-fractional partial differential equations in mathematical physics, Commun. Theor. Phys. (Beijing) 62 (2014), no. 5, 689-696. · Zbl 1304.35756 |
| [23] | S. Zhang, New exact solutions of the KdV-Burgers-Kuramoto equation, Phys. Lett. A 358 (2006), no. 5-6, 414-420. · Zbl 1142.35592 |
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