Exact solutions of generalized Boussinesq-Burgers equations and (2+1)-dimensional Davey-Stewartson equations. (English) Zbl 1267.35167

Summary: We study two coupled systems of nonlinear partial differential equations, namely, generalized Boussinesq-Burgers equations and (2+1)-dimensional Davey-Stewartson equations. The Lie symmetry method is utilized to obtain exact solutions of the generalized Boussinesq-Burgers equations. The travelling wave hypothesis approach is used to find exact solutions of the (2+1)-dimensional Davey-Stewartson equations.


35Q35 PDEs in connection with fluid mechanics
35B06 Symmetries, invariants, etc. in context of PDEs
35C07 Traveling wave solutions
Full Text: DOI


[1] M. Wang, Y. Zhou, and Z. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters A, vol. 216, no. 1-5, pp. 67-75, 1996. · Zbl 1125.35401
[2] J. Hu, “Explicit solutions to three nonlinear physical models,” Physics Letters A, vol. 287, no. 1-2, pp. 81-89, 2001. · Zbl 0971.34001
[3] J. Hu and H. Zhang, “A new method for finding exact traveling wave solutions to nonlinear partial differential equations,” Physics Letters A, vol. 286, no. 2-3, pp. 175-179, 2001. · Zbl 0969.35532
[4] S.-Y. Lou and J. Lu, “Special solutions from the variable separation approach: the Davey-Stewartson equation,” Journal of Physics A, vol. 29, no. 14, pp. 4209-4215, 1996. · Zbl 0899.35101
[5] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0762.35001
[6] C. H. Gu, Soliton Theory and Its Application, Zhejiang Science and Technology Press, Zhejiang, China, 1990.
[7] V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer Series in Nonlinear Dynamics, Springer, Berlin, Germany, 1991. · Zbl 0744.35045
[8] R. Hirota, The Direct Method in Soliton Theory, vol. 155 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 2004. · Zbl 1099.35111
[9] E. M. E. Zayed and K. A. Gepreel, “The (G’/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics,” Journal of Mathematical Physics, vol. 50, no. 1, Article ID 013502, 2009.
[10] Z. Yan, “A reduction mKdV method with symbolic computation to construct new doubly-periodic solutions for nonlinear wave equations,” International Journal of Modern Physics C, vol. 14, no. 5, pp. 661-672, 2003. · Zbl 1082.35518
[11] Z. Yan, “The new tri-function method to multiple exact solutions of nonlinear wave equations,” Physica Scripta, vol. 78, no. 3, Article ID 035001, 2008. · Zbl 1155.35427
[12] Z. Yan, “Periodic, solitary and rational wave solutions of the 3D extended quantum Zakharov-Kuznetsov equation in dense quantum plasmas,” Physics Letters A, vol. 373, no. 29, pp. 2432-2437, 2009. · Zbl 1231.76362
[13] D. Lu and B. Hong, “New exact solutions for the (2+1)-dimensional generalized Broer-Kaup system,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 572-580, 2008. · Zbl 1138.76023
[14] A.-M. Wazwaz, “The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1179-1195, 2005. · Zbl 1082.65584
[15] Z. Yan and H. Zhang, “New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water,” Physics Letters A, vol. 285, no. 5-6, pp. 355-362, 2001. · Zbl 0969.76518
[16] D. Lü, “Jacobi elliptic function solutions for two variant Boussinesq equations,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1373-1385, 2005. · Zbl 1072.35567
[17] Z. Yan, “Abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional integrable Davey-Stewartson-type equation via a new method,” Chaos, Solitons and Fractals, vol. 18, no. 2, pp. 299-309, 2003. · Zbl 1069.37060
[18] M. Wang and X. Li, “Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations,” Physics Letters A, vol. 343, no. 1-3, pp. 48-54, 2005. · Zbl 1181.35255
[19] J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 30, no. 3, pp. 700-708, 2006. · Zbl 1141.35448
[20] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. · Zbl 0698.35001
[21] P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1993. · Zbl 0785.58003
[22] N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1-3, CRC Press, Boca Raton, Fla, USA, 1994. · Zbl 0864.35001
[23] L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, NY, USA, 1982. · Zbl 0485.58002
[24] A. R. Adem and C. M. Khalique, “Symmetry reductions, exact solutions and coservation laws of a new coupled KdV system,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, pp. 3465-3475, 2012. · Zbl 1248.35180
[25] P. Wang, B. Tian, W.-J. Liu, X. Lü, and Y. Jiang, “Lax pair, Bäcklund transformation and multi-soliton solutions for the Boussinesq-Burgers equations from shallow water waves,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1726-1734, 2011. · Zbl 1433.35302
[26] A. Davey and K. Stewartson, “On three-dimensional packets of surface waves,” Proceedings of the Royal Society A, vol. 338, pp. 101-110, 1974. · Zbl 0282.76008
[27] S. A. El-Wakil, M. A. Abdou, and A. Elhanbaly, “New solitons and periodic wave solutions for nonlinear evolution equations,” Physics Letters A, vol. 353, no. 1, pp. 40-47, 2006. · Zbl 1106.65115
[28] E. Fan and J. Zhang, “Applications of the Jacobi elliptic function method to special-type nonlinear equations,” Physics Letters A, vol. 305, no. 6, pp. 383-392, 2002. · Zbl 1005.35063
[29] A. Bekir and A. C. Cevikel, “New solitons and periodic solutions for nonlinear physical models in mathematical physics,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 3275-3285, 2010. · Zbl 1196.35178
[30] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, NY, USA, 7th edition, 2007. · Zbl 1208.65001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.