×

Exact solutions for a third-order KdV equation with variable coefficients and forcing term. (English) Zbl 1188.35168

Summary: The general projective Riccati equation method and the exp-function method are used to construct generalized soliton solutions and periodic solutions to special KdV equation with variable coefficients and forcing term.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] N. Nirmala, M. J. Vedan, and B. V. Baby, “Auto-Bäcklund transformation, Lax pairs, and Painlevé property of a variable coefficient Korteweg-de Vries equation,” Journal of Mathematical Physics, vol. 27, no. 11, pp. 2640-2646, 1986. · Zbl 0632.35061 · doi:10.1063/1.527282
[2] E. Fan and Y. C. Hon, “Generalized tanh method extended to special types of nonlinear equations,” Zeitschrift fur Naturforschung A, vol. 57, no. 8, pp. 692-700, 2002.
[3] C. A. Gómez, “Exact solutions for a new fifth-order integrable system,” Revista Colombiana de Matemáticas, vol. 40, no. 2, pp. 119-125, 2006. · Zbl 1189.35274
[4] C. A. Gómez and A. H. Salas, “Exact solutions for a reaction diffusion equation by using the generalized tanh method,” Scientia et Technica, vol. 13, no. 35, pp. 409-410, 2007.
[5] C. A. Gómez, A. H. Salas, and B. Acevedo Frias, “New periodic and soliton solutions for the generalized BBM and Burger/s-BBM equations,” Applied Mathematics and Computation. In press. · Zbl 1203.35221 · doi:10.1016/j.amc.2009.05.068
[6] C. A. Gómez and A. H. Salas, “Exact solutions for a new integrable system (KdV6),” Journal of Mathematical Sciences: Advances and Applications, vol. 1, no. 2, pp. 401-413, 2008. · Zbl 1182.35064
[7] A. H. Salas, C. A. Gómez, and G. Escobar, “Exact solutions for the general fifth order KdV equation by the extended tanh method,” Journal of Mathematical Sciences: Advances and Applications, vol. 1, no. 2, pp. 305-310, 2008. · Zbl 1179.65132
[8] C. A. Gómez, “A new travelling wave solution of the Mikhailov-Novikov-Wang system using the extended tanh method,” Boletín de Matemáticas, vol. 14, no. 1, pp. 38-43, 2007. · Zbl 1203.35219
[9] A.-M. Wazwaz, “The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 1002-1014, 2007. · Zbl 1115.65106 · doi:10.1016/j.amc.2006.07.002
[10] C. A. Gómez, “Special forms of the fifth-order KdV equation with new periodic and soliton solutions,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1066-1077, 2007. · Zbl 1122.65393 · doi:10.1016/j.amc.2006.11.158
[11] C. A. Gómez and A. H. Salas, “The generalized tanh-coth method to special types of the fifth-order KdV equation,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 873-880, 2008. · Zbl 1154.65364 · doi:10.1016/j.amc.2008.05.105
[12] A. H. Salas and C. A. Gómez, “Computing exact solutions for some fifth KdV equations with forcing term,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 257-260, 2008. · Zbl 1160.35526 · doi:10.1016/j.amc.2008.06.033
[13] C. A. Gómez and A. H. Salas, “The variational iteration method combined with improved generalized tanh-coth method applied to Sawada-Kotera equation,” Applied Mathematics and Computation. In press. · Zbl 1203.65196 · doi:10.1016/j.amc.2009.05.046
[14] 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 · doi:10.1016/j.chaos.2006.03.020
[15] S. Zhang, “Exp-function method exactly solving the KdV equation with forcing term,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 128-134, 2008. · Zbl 1135.65388 · doi:10.1016/j.amc.2007.07.041
[16] J.-H. He and L.-N. Zhang, “Generalized solitary solution and compacton-like solution of the Jaulent-Miodek equations using the Exp-function method,” Physics Letters A, vol. 372, no. 7, pp. 1044-1047, 2008. · Zbl 1217.35152 · doi:10.1016/j.physleta.2007.08.059
[17] S. Zhang, “Application of Exp-function method to a KdV equation with variable coefficients,” Physics Letters A, vol. 365, no. 5-6, pp. 448-453, 2007. · Zbl 1203.35255 · doi:10.1016/j.physleta.2007.02.004
[18] A. H. Salas, “Exact solutions for the general fifth KdV equation by the exp function method,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 291-297, 2008. · Zbl 1160.35525 · doi:10.1016/j.amc.2008.07.013
[19] A. H. Salas, C. A. Gómez, and J. Castillo, “New abundant solutions for the Burger/s equation,” Computers & Mathematics with Applications, vol. 58, pp. 514-520, 2009. · Zbl 1189.35289
[20] C. A. Gómez and A. H. Salas, “The Cole-Hopf transformation and improved tanh-coth method applied to new integrable system (KdV6),” Applied Mathematics and Computation, vol. 204, no. 2, pp. 957-962, 2008. · Zbl 1157.65457 · doi:10.1016/j.amc.2008.08.006
[21] R. Conte and M. Musette, “Link between solitary waves and projective Riccati equations,” Journal of Physics. A, vol. 25, no. 21, pp. 5609-5623, 1992. · Zbl 0782.35065 · doi:10.1088/0305-4470/25/21/019
[22] A. H. Salas, “Some solutions for a type of generalized Sawada-Kotera equation,” Applied Mathematics and Computation, vol. 196, no. 2, pp. 812-817, 2008. · Zbl 1132.35461 · doi:10.1016/j.amc.2007.07.013
[23] Z. Yan, “The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equations,” MMRC, AMSS, Academis Sinica, vol. 22, pp. 275-284, 2003.
[24] E. Yomba, “The general projective Riccati equations method and exact solutions for a class of nonlinear partial differential equations,” Chinese Journal of Physics, vol. 43, no. 6, pp. 991-1003, 2005.
[25] C. A. Gómez and A. H. Salas, “Special forms of Sawada-Kotera equation with periodic and soliton solutions,” International Journal of Applied Mathematical Analysis and Applications, vol. 3, no. 1, pp. 45-51, 2008. · Zbl 1266.35128
[26] Y. Shang, Y. Huang, and W. Yuan, “New exact traveling wave solutions for the Klein-Gordon-Zakharov equations,” Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1441-1450, 2008. · Zbl 1155.35443 · doi:10.1016/j.camwa.2007.10.033
[27] C. A. Gómez and A. H. Salas, “Exact solutions for the generalized shallow water wave equation by the general projective Riccati equations method,” Boletín de Matemáticas, vol. 13, no. 1, pp. 50-56, 2006. · Zbl 1203.35220
[28] C. A. Gómez and A. H. Salas, “New exact solutions for the combined sinh-cosh-Gordon equation,” Lecturas Matematicas, vol. 27, pp. 87-93, 2006. · Zbl 1381.35009
[29] C. A. Gómez, “New exact solutions of the Mikhailov-Novikov-Wang system,” International Journal of Computer, Mathematical Sciences and Applications, vol. 1, pp. 137-143, 2007.
[30] Y. Chen and B. Li, “General projective Riccati equation method and exact solutions for generalized KdV-type and KdV-Burgers-type equations with nonlinear terms of any order,” Chaos, Solitons & Fractals, vol. 19, no. 4, pp. 977-984, 2004. · Zbl 1057.35051 · doi:10.1016/S0960-0779(03)00250-9
[31] Taogetusang and Sirendaoerji, “The Jacobi elliptic function-like exact solutions to two kinds of KdV equations with variable coefficients and KdV equation with forcible term,” Chinese Physics, vol. 15, no. 12, pp. 2809-2818, 2006. · doi:10.1088/1009-1963/15/12/008
[32] S.-D. Zhu, “Exp-function method for the Hybrid-Lattice system,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, pp. 461-464, 2007. · Zbl 06942293
[33] S.-D. Zhu, “Exp-function method for the discrete mKdV lattice,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, pp. 465-468, 2007. · Zbl 06942294
[34] C. Dai, X. Cen, and S. S. Wu, “The application of He/s exp-function method to a nonlinear differential-difference equation,” Chaos, Solitons & Fractals, vol. 41, no. 1, pp. 511-515, 2009. · Zbl 1198.65136 · doi:10.1016/j.chaos.2008.02.021
[35] C.-Q. Dai and J.-F. Zhang, “Application of he/s EXP-function method to the stochastic mKdV equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 5, pp. 675-680, 2009. · Zbl 06942438
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.