Explicit Solutions of a Class of (3+1)-Dimensional Nonlinear Model

Year 2018, , 184 - 190, 25.12.2018

Yongyi Gu

https://doi.org/10.33401/fujma.486881

Abstract

In this article, we employ Lie group analysis to obtain symmetry reduction of a class of (3+1)-dimensional nonlinear model. This nonlinear model plays a critical role in the study of nonlinear sciences. By the exp$(-\varphi(z))$-expansion method, we construct explicit solutions for the proposed equation. Four types of explicit solutions are obtained, which are hyperbolic, exponential, trigonometric and rational function solutions.

Keywords

Differential equation, Explicit solutions, Symmetry reduction, Exp$(-\varphi(z))$-expansion method

References

• [1] J. Q. Mei, H. Q. Zhang, New soliton-like and periodic-like solutions for the KdV equation, Appl. Math. Comput., 169 (2005), 589-599.
• [2] A. G. Cui, H. Y. Li, C. Y. Zhang, A splitting method for shifted skew-Hermitian linear system, J. Inequal. Appl., 2016, 160 (2016); doi: 10.1186/s13660- 016-1105-1.
• [3] V. E. Zakharov, E. A. Kuznetsov, On three-dimensional solitons, Sov. Phys. JETP 39 (1974), 285-288.
• [4] D. J. Zhang, The N-soliton solutions for the modified KdV equation with self-consistent Sources, J. Phys. Soc. Japan, 71(11) (2002), 2649-2656.
• [5] A. M. Wazwaz, New solitons and kink solutions for the Gardner equation, Commun. Nonlinear Sci. Numer. Simul., 12 (2007), 1395-1404.
• [6] F. Tascan, A. Bekir, M. Koparan, Travelling wave solutions of nonlinear evolution equations by using the first integral method, Commun. Nonlinear Sci. Numer. Simul., 14(5) (2009), 1810-1815.
• [7] S. Tang, Y. Xiao, Z. Wang, Travelling wave solutions for a class of nonlinear fourth order variant of a generalized Camassa-Holm equation, Appl. Math. Comput., 210(1) (2009), 39-47.
• [8] A. J. M. Jawad, M. D. Petkovic, A. Biswas, Modified simple equation method for nonlinear evolution equations, Appl. Math. Comput., 217(2) (2010), 869-877.
• [9] D. Feng, T. He, J. L¨u, Bifurcations of travelling wave solutions for (2+1)-dimensional Boussinesq type equation, Appl. Math. Comput., 185(1) (2007), 402-414.
• [10] K. Khan, M. A. Akbar, Study of analytical method to seek for exact solutions of variant Boussinesq equations, Springerplus, 3 (2014), 17 pages.
• [11] W. J. Yuan, Y. Z. Li, J. M. Lin, Meromorphic solutions of an auxiliary ordinary differential equation using complex method, Math. Meth. Appl. Sci., 36 (2013), 1776-1782.
• [12] W. J. Yuan, F. N. Meng, Y. Huang, Y. H. Wu, All traveling wave exact solutions of the variant Boussinesq equations, Appl. Math. Comput., 268 (2015), 865-872.
• [13] Y. Y. Gu, W. J. Yuan, N. Aminakbari, Q. H. Jiang, Exact solutions of the Vakhnenko-Parkes equation with complex method, J. Funct. Spaces, (2017), Article ID 6521357, 6 pages.
• [14] Y. Y. Gu, N. Aminakbari, W. J. Yuan, Y. H. Wu, Meromorphic solutions of a class of algebraic differential equations related to Painleve equation III, Houston J. Math., 43(4) (2017), 1045-1055.
• [15] Y. Y. Gu, W. J. Yuan, N. Aminakbari, J. M. Lin, Meromorphic solutions of some algebraic differential equations related Painleve equation IV and its applications, Math. Meth. Appl. Sci., 41(10) (2018), 3832-3840.
• [16] K. Khan, M. A. Akbar, The exp$(-\phi(\xi))$-expansion method for finding travelling wave solutions of Vakhnenko-Parkes equation, Int. J. Dyn. Syst. Diff K. er. Equ., 5(1) (2014), 72-83.
• [17] S. M. R. Islam, K. Khan, M. A. Akbar, Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations, Springerplus, 4 (2015), 11 pages.
• [18] N. Kadkhoda, H. Jafari, Analytical solutions of the Gerdjikov-Ivanov equation by using exp$(-\varphi(\xi))$-expansion method, Optik, 139 (2017), 72-76.
• [19] C. Tian, Lie group and its applications in partial differential equations, Higher Education Press, Beijing, 2001.
• [20] H. Liu, J. Li, Q. Zhang, Lie symmetry analysis and exact explicit solutions for gener Burgers’ equation, J. Comput. Appl. Math., 228(1) (2009), 1-9.
• [21] B. Ren, Symmetry reduction related with nonlocal symmetry for Gardner equation, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 456-463.
• [22] H. Djourdem, S. Benaicha, Solvability for a nonlinear third-order three-point boundary value problem, Univers. J. Math. Appl., 1(2) (2018), 125-131.
• [23] İ. Çelik, Chebyshev Wavelet collocation method for solving a class of linear and nonlinear nonlocal boundary value problems, Fundam. J. Math. Appl., 1(1) (2018), 25-35.
• [24] B. Noureddine, S. Benaicha, Existence of solutions for nonlocal boundary value problem for Caputo nonlinear fractional differential inclusion, J. Math. Sci. Model. 1(1) (2018), 45-55.
• [25] B. Noureddine, S. Benaicha, H. Djourdem, Positive solutions for nonlinear fractional differential equation with nonlocal boundary conditions, Univers. J. Math. Appl. 1(1) (2018), 39-45.