In the present article, we construct the exact traveling wave solutions of nonlinear PDEs in mathematical physics via the variant Boussinesq equations and the coupled KdV equations by using the extended mapping method and auxiliary equation method. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.
M. J. Ablowitz and P. A. Clarkson, “Soliton, Nonlinear Evolution Equations and Inverse Scattering,” Cambridge University Press, Cambridge, 1991. doi:10.1017/CBO9780511623998
C. H. Gu, et al., “Soliton Theory and Its Application,” Zhejiang Science and Technology Press, Zhejiang, 1990.
V. B. Matveev and M. A. Salle, “Darboux Transformation and Soliton,” Springer, Berlin, 1991.
R. Hirota, “The Direct Method in Soliton Theory,” Cambridge University Press, Cambridge, 2004. doi:10.1017/CBO9780511543043
S. Y. Lou and J. Z. Lu, “Special Solutions from Variable Separation Approach: Davey-Stewartson Equation,” Journal of Physics A: Mathematical and General, Vol. 29, No. 14, 1996, pp. 4209-4215. doi:10.1088/0305-4470/29/14/038
E. J. Parkes and B. R. Duffy, “Travelling Solitary Wave Solutions to a Compound KdV-Burgers Equation,” Physics Letters A, Vol. 229, No. 4, 1997, pp. 217-220. doi:10.1016/S0375-9601(97)00193-X
E. Fan, “Extended Tanh-Function Method and Its Applications to Nonlinear Equations,” Physics Letters A, Vol. 277, No. 4-5, 2000, pp. 212-218. doi:10.1016/S0375-9601(00)00725-8
Z. Y. Yan, “New Explicit Travelling Wave Solutions for Two New Integrable Coupled Nonlinear Evolution Equations,” Physics Letters A, Vol. 292, No. 1-2, 2001, pp. 100-106. doi:10.1016/S0375-9601(01)00772-1
Y. Chen and Y. Zheng, “Generalized Extended TanhFunction Method to Construct New Explicit Exact Solutions for the Approximate Equations for Long Water Waves,” International Journal of Modern Physics C, Vol. 14, No. 5, 2003, pp. 601-611. doi:10.1142/S0129183103004760
M. L. Wang, “Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Equations in Mathematical Physics,” Physics Letters A, Vol. 216, No. 1-5, 1996, pp. 67-75. doi:10.1016/0375-9601(96)00283-6
G. W. Bluman and S. Kumei, “Symmetries and Differential Equations,” Springer-Verlag, New York, 1989.
P. J. Olver, “Applications of Lie Groups to Differential Equations,” Springer-Verlag, New York, 1986. doi:10.1007/978-1-4684-0274-2
E. M. E. Zayed and K. A. Gepreel, “The -Expansion Method for Finding Traveling Wave Solutions of Nonlinear PDEs in Mathematical Physics,” Journal of Mathematical Physics, Vol. 50, No. 1, 2009, 013502. doi:10.1063/1.3033750
Z. Y. Yan, “A Reduction mKdV Method with Symbolic Computation to Constract New Doubly-Periodic Solutions for Nonlinear Wave Equations,” International Journal of Modern Physics C, Vol. 14, No. 5, 2003, pp. 661-672. doi:10.1142/S0129183103004814
Z. Y. Yan, “The New Tri-Function Method to Multiple Exact Solutions of Nonlinear Wave Equations,” Physica Scripta, Vol. 78, No. 3, 2008, 035001. doi:10.1088/0031-8949/78/03/035001
Z. Y. 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, 2009, pp. 2432-2437. doi:10.1016/j.physleta.2009.04.018
D. C. Lu and B. J. Hong, “New Exact Solutions for the (2+1)-Dimensional Generalized Broer-Kaup System,” Applied Mathematics and Computation, Vol. 199, No. 2, 2008, pp. 572-580. doi:10.1016/j.amc.2007.10.012
A. V. Porubov, “Periodical Solution to the Nonlinear Dissipative Equation for Surface Waves in a Convecting Liquid,” Physics Letters A, Vol. 221, No. 6, 1996, pp. 391-394. doi:10.1016/0375-9601(96)00598-1
M. Wazwaz, “The Tanh and Sine-Cosine Method for Compact and Noncompact Solutions of Nonlinear Klein Gordon Equation,” Applied Mathematics and Computation, Vol. 167, No. 2, 2005, pp. 1179-1195. doi:10.1016/j.amc.2004.08.006
Z. Y. Yan and H. Q. Zhang, “New Explicit Solitary Wave Solutions and Periodic Wave Solutions for WhithamBroer-Kaup Equation in Shallow Water,” Physics Letters A, Vol. 285, No. 5-6, 2001, pp. 355-362. doi:10.1016/S0375-9601(01)00376-0
D. C. Lu, “Jacobi Elliptic Functions Solutions for Two Variant Boussinesq Equations,” Chaos, Solitons and Fractals, Vol. 24, No. 5, 2005, pp. 1373-1385. doi:10.1016/j.chaos.2004.09.085
Z. Y. Yan, “Abundant Families of Jacobi Elliptic Functions of the (2+1) Dimensional Integrable Davey-Stawartson-Type Equation via a New Method,” Chaos, Solitons and Fractals, Vol. 18, No. 2, 2003, pp. 299-309. doi:10.1016/S0960-0779(02)00653-7
C. L. Bai and H. Zhao, “Generalized Method to Construct the Solitonic Solutions to (3+1)-Dimensional Nonlinear Equation,” Physics Letters A, Vol. 354, No. 5-6, 2006, pp. 428-436. doi:10.1016/j.physleta.2006.01.084
F. Cariello and M. Tabor, “Similarity Reductions from Extended Painleve’ Expansions for Nonintegrable Evolution Equations,” Physica D: Nonlinear Phenomena, Vol. 53, No. 1, 1991, pp. 59-70. doi:10.1016/0167-2789(91)90164-5
M. Wang and X. Li, “Extended F-Expansion and Periodic Wave Solutions for the Generalized Zakharov Equations,” Physics Letters A, Vol. 343, No. 1-3, 2005, pp. 4854. doi:10.1016/j.physleta.2005.05.085
X. Feng, “Exploratory Approach to Explicit Solution of Nonlinear Evolution Equations,” International Journal of Theoretical Physics, Vol. 39, 2000, p. 222. doi:10.1023/A:1003615705115
J. L. Hu, “Explicit Solutions to Three Nonlinear Physical Models,” Physics Letters A, Vol. 287, No. 1-2, 2001, pp. 81-89. doi:10.1016/S0375-9601(01)00461-3
J. L. Hu, “A New Method for Finding Exact Traveling Wave Solutions to Nonlinear Partial Differential Equations,” Physics Letters A, Vol. 286, No. 2-3, 2001, pp. 175-179. doi:10.1016/S0375-9601(01)00291-2
J. H. He and X. H.Wu, “Exp-Function Method for Nonlinear Wave Equations,” Chaos, Solitons and Fractals, Vol. 30, No. 3, 2006, pp. 700-708. doi:10.1016/j.chaos.2006.03.020
X. Z. Li and M. L. Wang, “A Sub-ODE Method for Finding Exact Solutions of a Generalized KdV-mKdV Equation with Higher Order Nonlinear Terms,” Physics Letters A, Vol. 361, No. 1-2, 2007, pp. 115-118. doi:10.1016/j.physleta.2006.09.022
E. M. E. Zayed and K. A. Gepreel, “Three Types of TravelingWave Solutions for Nonlinear Evolution Equations Using the -Expansion Method,” International Journal of Nonlinear Science, Vol. 7, 2009, p. 501.
M. L. Wang, “Solitary Wave Solutions for Variant Boussinesq Equations,” Physics Letters A, Vol. 199, No. 3-4, 1995, pp. 169-172. doi:10.1016/0375-9601(95)00092-H
E. G. Fan and Y. C. Hon, “A Series of Travelling Wave Solutions for Two Variant Boussinesq Equations in Shallow Water Waves,” Chaos, Solitons and Fractals, Vol. 15, No. 3, 2003, pp. 559-566. doi:10.1016/S0960-0779(02)00144-3
R. Hirota and J. Satsuma, “Soliton Solutions of Coupled Korteweg-de Vries Equation,” Physics Letters A, Vol. 85, No. 8-9, 1981, pp. 407-408. doi:10.1016/0375-9601(81)90423-0
K. A. Gepreel, “Exact Solutions for Nonlinear PDEs with the Variable Coefficients in Mathematical Physics,” Journal of Information and Computing Science, Vol. 6, No. 1, 2011, pp. 3-14.