Pure and Applied Mathematics Journal
Volume 5, Issue 6, December 2016, Pages: 174-180
Received: Sep. 2, 2016;
Accepted: Sep. 18, 2016;
Published: Oct. 17, 2016
Views 3111 Downloads 167
Baodan Tian, School of science, Southwest University of Science and Technology, Mianyang, China
Yanhong Qiu, School of science, Southwest University of Science and Technology, Mianyang, China
In this paper, a simple direct method is presented to find equivalence transformation of a nonlinear Whitham- Broer-Kaup equations. Applying this equivalence transformation, we can obtain the symmetry group theorem of the Whitham-Broer-Kaup equations and then derive series of new exact and explicit solutions of the Whitham-Broer-Kaup equations according to solutions of the previous references.
Exact and Explicit Solutions of Whitham-Broer-Kaup Equations in Shallow Water, Pure and Applied Mathematics Journal.
Vol. 5, No. 6,
2016, pp. 174-180.
M. J Ablowitz, Soliton, Nolinear Evolution Equations and Inverse Scatting, Cambridge University Press, New York, 1991.
L. J. F. Broer, Approximate equations for long water waves, Applied Scientific Research 31 (1975) 377–395.
Y. Chen, Q. Wang, B. Li, A generalized method and general form solutions to the Whitham-Broer-Kaup equation, Chaos, Solitons & Fractals, 22(2004)675-682.
Y. Chen, Q. Wang, Multiple Riccati equations rational expansion method and complexions solutions of the Whitham-Broer-Kaup equation, Physics Letters A, 347 (2005)215-227.
Y. Chen, Q. Wang, B. Li, Elliptic equation rational expansion method and new exact travelling solutions for Whitham-Broer-Kaup equations, Chaos, Solitons & Fractals, 26(2005)231-246.
P. A. Clarkson, M. D. Kruska, New similarity solutions of the Boussinesq equation, Journal of Mathematical Physics, 30 (1989) 2201-2213.
S. M. El-Sayed, D. Kaya, Exact and numerical traveling wave solutions of Whitham-Broer-Kaup equations, Applied Mathe- matics and Computation,167(2005)1339-1349.
E. G. Fan, H. Q. Zhang, Backlund transformation and exact solutions for Whitham-Broer-Kaup Equations in shallow water, Applied Mathematics and Mechanics, 19(1998) 667-670.
X. Y. Jiao, H. Q. Zhang, An extended method and its application to Whitham-Broer-Kaup equation and two- dimensional perturbed KdV equation, Applied Mathematics and Computation, 172(2006)664-677.
D. J. Kaup, A higher-order water-wave equation and the method for solving it, Progress of Theoretical Physics 54 (1975) 396–408.
S. Y. Lou, H. C. Ma, Non-Lie symmetry groups of (2+1)- dimensional nonlinear systems obtained from a simple direct method, Journal of Physics A: Mathematical and General, 38 (2005)129-137.
H. C. Ma, A simple method to generate Lie point symmetry groups of the (3+1)-dimensional Jimbo-Miwa equation, Chinese Physics Letters, 22(2005)554-557.
P. J. Olver, Applications of Lie groups to differential equations, Springer-Verlag, 1993.
J. W. Shen, W. Xu, Y. F. Jin, Bifurcation method and traveling wave solution to Whitham-Broer-Kaup equation, Applied Mathematics and Computation,171(2005)677-702.
Sirendaoreji, A new auxiliary equation and exact travelling wave solutions of nonlinear equations, Physics Letters A, 356 (2006)124-130.
G. B. Whitham, Variational methods and applications to water waves, Proceedings of the Royal Society of London, Series A 299 (1967) 6–25.
F. D. Xie, Z. Y. Yan, H. Q Zhang, Explicit and exact traveling wave solutions of Whitham-Broer-Kaup shallow water equations, Physics Letters A, 285(2001)76-80.
G. Q. Xu, An elliptic equation method and its applications in nonlinear evolution equations, Chaos, Solitons & Fractals, 29 (2006)942-947.
G. Q. Xu, Z. B. Li, Exact travelling wave solutions of the Whitham-Broer-Kaup and Broer–Kaup–Kupershmidt equations, Chaos, Solitons & Fractals, 24(2005)549-556.
T. Xu, J. Li, H. Q. Zhang,et al, New extension of the tanh- function method and application to Whitham-Broer-Kaup shallow water model with symbolic computation, Physics Letters A, 369 (2007) 458-463.
Z. Y. Yan, H. Q Zhang, New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water, Physics Letters A, 285(2001) 355-362.