×
Hong, Baojian; Lu, Dianchen

New Jacobi elliptic function-like solutions for the general KdV equation with variable coefficients. (English) Zbl 1255.35193

Math. Comput. Modelling 55, No. 3-4, 1594-1600 (2012).
Summary: An improved general mapping deformation method based on the generalized Jacobi elliptic functions expansion method with computerized symbolic computation is used to construct more new exact solutions of a generalized KdV equation with variable coefficients. As a result, seven families of new generalized Jacobi elliptic function-like solutions, soliton-like solutions and trigonometric function solutions of the equation are obtained by using this method, which shows that the general 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.

Cited in 7 Documents

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35C08 Soliton solutions
33E05 Elliptic functions and integrals
35B10 Periodic solutions to PDEs

Keywords:

generalized KdV equation with variable coefficients; improved general mapping deformation method; exact solutions; generalized Jacobi elliptic function-like solutions
PDF BibTeX XML Cite
\textit{B. Hong} and \textit{D. Lu}, Math. Comput. Modelling 55, No. 3--4, 1594--1600 (2012; Zbl 1255.35193)
Full Text: DOI

References:

[1] Ablowitz, M. J.; Clarkson, P. A., Soliton, Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge University Press: Cambridge University Press New York · Zbl 0762.35001
[2] Gardner, C. S.; Greene, J. M.; Kruskal, M. D., Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19, 1095-1097 (1967) · Zbl 1103.35360
[3] Lu, D. C.; Hong, B. J., Bäcklund transformation and \(n\)-soliton-like solutions to the combined KdV-Burgers equation with variable coefficients, Int. J. Nonlinear Sci., 1, 2, 3-10 (2006) · Zbl 1394.35113
[4] Hu, H. C.; Tang, X. Y.; Lou, S. Y.; Liu, Q. P., Variable separation solutions obtained from Darboux Transformations for the asymmetric Nizhnik-Novikov-Veselov system, Chaos Solitons Fractals, 22, 2, 327-334 (2004) · Zbl 1063.35135
[5] He, J. H., Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. Numer. Simul., 6, 207-208 (2005) · Zbl 1401.65085
[6] Lu, D. C.; Hong, B. J., New exact solutions for the (2+1)-dimensional Generalized Broer-Kaup System, Appl. Math. Comput., 199, 2, 572-580 (2008) · Zbl 1138.76023
[7] He, J. H., Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B, 20, 1141-1199 (2006) · Zbl 1102.34039
[8] Abdusalam, H. A., On an improved complex tanh-function method, Int. J. Nonlinear Sci. Numer. Simul., 6, 99-106 (2005) · Zbl 1401.35012
[9] Hu, J. Q., An algebraic method exactly solving two highdimensional nonlinear evolution equations, Chaos Solitons Fractals, 23, 391-398 (2005) · Zbl 1069.35065
[10] Huang, W. H.; Liu, Y. L., Jacobi elliptic function solutions of the Ablowitz-Ladik discrete nonlinear Schrodinger system, Chaos Solitons Fractals, 40, 786-792 (2009) · Zbl 1197.81121
[11] Feng, Y.; Zhang, H. Q., A new auxiliary function method for solving the generalized coupled Hirota-Satsuma KdV system, Appl. Math. Comput., 200, 283-288 (2008) · Zbl 1143.65081
[12] Zayed, E. M.E.; Gepreel, Khaled A., Some applications of the G’/G-expansion method to non-linear partial differential equations, Appl. Math. Comput., 212, 1-13 (2009) · Zbl 1166.65386
[13] Hong, B. J., New Jacobi elliptic functions solutions for the variable-coefficient mKdV equation, Appl. Math. Comput., 2158, 2908-2913 (2009) · Zbl 1180.35459
[14] Hong, B. J., New exact Jacobi elliptic functions solutions for the generalized coupled Hirota-Satsuma KdV system, Appl. Math. Comput., 217, 2, 472-479 (2010) · Zbl 1200.35260
[15] Mishra, Ajay; Kumar, Ranjit, Exact solutions of variable coefficient nonlinear diffusion-reaction equations with a nonlinear convective term, Phys. Lett. A, 374, 29, 2921-2924 (2010) · Zbl 1237.35096
[16] Zhao, X. Q.; Tang, D. B.; Wang, L. M., New soliton-like solutions for KdV equation with variable coefficient, Phys. Lett. A, 346, 288-291 (2005) · Zbl 1195.35275
[17] Zhu, J. M.; Zheng, C. L.; Ma, Z. Y., A general mapping approach and new travelling wave solutions to the general variable coefficient KdV equation, Chin. Phys., 13, 12, 2008-2012 (2004)
[18] Fu, Z. T.; Liu, S. D.; Liu, S. K.; Zhao, Q., New exact solutions to KdV equations with variable coefficients or forcing, Appl. Math. Mech. (English Ed.), 25, 73-79 (2004) · Zbl 1061.35109
[19] Zhang, J. F.; Cheng, F. Y., Truncated expansion method and new exact solution-like solution of the general variable coefficient KdV equation, Acta Phys. Sinica, 50, 9, 1648-1650 (2001) · Zbl 1202.35283
[20] Li, D. S.; Zhang, H. Q., Improved tanh-function method and the new exact solutions for the general variable coefficient KdV equation and MKdV equation, Acta Phys. Sinica, 52, 7, 1569-1573 (2003) · Zbl 1202.81044
[21] Chan, W. L.; Li, K. S., Nonpropagating solitons of the variable coefficient and nonisospectral Korteweg-de Vries equation, J. Math. Phys., 30, 11, 2521-2526 (1989) · Zbl 0698.35141
[22] Chou, T., Symmetries and a hierarchy of the general KdV equation, J. Phys. A: Math. Gen., 20, 359-366 (1987)
[23] Lou, S. Y.; Ruan, H. Y., Conservation laws of the variable coefficient KdV and MKdV equations, Acta Phys. Sinica, 41, 2, 182-187 (1992)
[24] Chan, W. L.; Zhang, X., Symmetries, conservation laws and Hamiltonian structures of the non-isospectral and variable coefficient KdV and MKdV equations, J. Phys. A: Math. Gen., 28, 407-412 (1995) · Zbl 0853.35105
[25] Fan, E. G., The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials, Phys. Lett. A, 375, 3, 493-497 (2011) · Zbl 1241.35176
[26] Zhang, J. F.; Han, P., Symmetries of the variable coefficient KdV equation and three hierarchies of the integrodifferential variable coefficient KdV equation, Chin. Phys. Lett., 11, 12, 721-723 (1994)
[27] Brugarino, T., Painleve property, auto-Bäcklund transformation, Lax pairs, and reduction to the standard form for the Korteweg de Vries equation with nonuniformities, J. Math. Phys., 30, 1013-1015 (1989) · Zbl 0678.58038
[28] Brugarino, T.; Pantano, P., The integration of Burgers and Korteweg-de Vries equations with nonuniformities, Phys. Lett. A., 80, 223-224 (1980)
[29] Biswas, Anjan, Solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coefficients, Nonlinear Dynam., 58, 345-348 (2009) · Zbl 1183.35241
[30] Johnpillai, A. G.; Khalique, C. M.; Biswas, Anjan, Exact solutions of KdV equation with time-dependent coefficients, Appl. Math. Comput., 216, 3114-3119 (2010) · Zbl 1195.35263
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.