Exact Solution and Conservation Laws for Fifth-Order Korteweg-de Vries Equation


With the aid of Mathematica, new exact travelling wave solutions for fifth-order KdV equation are obtained by using the solitary wave ansatz method and the Wu elimination method. The derivation of conservation laws for a fifth-order KdV equation is considered.

Share and Cite:

Al-Ali, E. (2013) Exact Solution and Conservation Laws for Fifth-Order Korteweg-de Vries Equation. Journal of Applied Mathematics and Physics, 1, 49-53. doi: 10.4236/jamp.2013.15007.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] M. J. Abolwitz and P. A. Clarkson, “Solitons, Nonlinear Evolution Equation and Inverse Scattering,” Cambridge University Press, Cambridge, 1991.
[2] C. Rogers and P. Wong, “On reciprocal B€ a Acklund Transformations of Inverse Scattering Schemes,” Physica Scripta, Vol. 30, 1984, pp. 10-14.
[3] A. H. Khater, D. K. Callebaut, A. A. Abdalla, A. R. Shehata and S. M. Sayed, “Backlund Transformations and Exact Solutions for Self-Dual SU(3) Yang-Mills Equations,” IL Nuovo Cimento B, Vol. 114, 1999, pp. 1-10.
[4] C. Qu, Y. Si and R. Liu, “On Affine Sawada-Kotera Equation,” Chaos, Solitons & Fractals, Vol. 15, No. 1, 2003, pp. 131-139.
[5] O. C. Wright, “The Darboux Transformation of Some Manakov Systems,” Applied Mathematics Letters, Vol. 16, No. 5, 2003, pp. 647-652.
[6] R. Hirota, “The Direct Method in Soliton Theory,” Cambridge University Press, Cambridge, 2004.
[7] A. H. Khater, D. K. Callebaut and S. M. Sayed, “Exact Solutions for Some Nonlinear Evolution Equations which Describe Pseudospherical Surfaces,” Journal of Computational and Applied Mathematics, Vol. 189, No. 1-2, 2006, pp. 387-411. http://dx.doi.org/10.1016/j.cam.2005.10.007
[8] S. K. Liu, Z. T. Fu and S. D. Liu, “Jacobi Elliptic Function Expansion Method and Periodic Wave Solutions of Nonlinear Wave Equations,” Physics Letters A, Vol. 289, No. 1-2, 2001, pp. 69-74.
[9] E. Fan, “Extended Tanh-Function Method and Its Applications to Nonlinear Equations,” Physics Letters A, Vol. 277, 2000, pp. 212-219.
[10] W. Malfliet and W. Hereman, “The Tanh Method I. Exact Solutions of Nonlinear Wave Equations,” Physica Scripta, Vol. 54, No. 6, 1996, pp. 569-575.
[11] K. Chadan and P. C. Sabatier, “Inverse Problem in Quantum Scattering Theory,” Springer, New York, 1977.
[12] M. J. Ablowitz, S. Chakravarty and R. Halburd, “On Painlevé and Darboux-Halphen Type Equations, in the Painlevé Property, One Century Later,” In: R. Conte, Ed., CRM Series in Mathematical Physics, Springer, Berlin, 1998.
[13] M. Elham Al-Ali, “Traveling Wave Solutions for a Generalized Kawahara and Hunter-Saxton Equations,” International Journal of Mathematical Analysis, Vol. 7, 2013, pp. 1647-1666.
[14] S. M. Sayed, “The B?cklund Transformations, Exact Solutions, and Conservation Laws for the Compound Modified Korteweg-de Vries-Sine-Gordon Equations which describe Pseudospherical Surfaces,” Journal of Applied Mathematics, Vol. 2013, 2013, pp. 1-7.
[15] V. B. Matveev and M. A. Salle, “Darboux Transformations and Solitons,” Springer-Verlag, Berlin, 1991.
[16] K. Tenenblat, “Transformations of Manifolds and Applications to Deferential Equations, Pitman Monographs and Surveys in Pure and Applied Mathematics 93,” Addison Wesley Longman, England, 1998.
[17] A. M. Wazwaz, “New Compactons, Solitons and Periodic Solutions for Nonlinear Variants of the KdV and the KP Equations,” Chaos, Solitons & Fractals, Vol. 22, 2004, pp. 249-260. http://dx.doi.org/10.1016/j.chaos.2004.01.005
[18] A. M. Wazwaz, “Two Reliable Methods for Solving Variants of the KdV Equation with Compact and Noncompact Structures,” Chaos, Solitons & Fractals, Vol. 28, No. 2, 2006, pp. 454-462.
[19] X. G. Geng and H. Wang, “Coupled Camassa-Holm Equations, N-Peakons and Infinitely Many Conservation Laws,” Journal of Mathematical Analysis and Applications, Vol. 403, 2013, pp. 262-271.
[20] A. H. Khater, D. K. Callebaut and S. M. Sayed, “Conservation Laws for Some Nonlinear Evolution Equations which Describe Pseudo-Spherical Surfaces,” Journal of Geometry and Physics, Vol. 51, No. 3, 2004, pp. 332-352.
[21] J. A. Cavalcante and K. Tenenblat, “Conservation Laws for Nonlinear Evolution Equations,” Journal of Mathematical Physics, Vol. 29, 1988, pp. 1044-1059.
[22] R. Beals, M. Rabelo and K. Tenenblat, “Backlund Transformations and Inverse Scattering Solutions for Some Pseudo-Spherical Surfaces,” Studies in Applied Mathematics, Vol. 81, 1989, pp. 125-134.
[23] E. G. Reyes, “Conservation Laws and Calapso-Guichard Deformations of Equations Describing Pseudo-Spherical Surfaces,” Journal of Mathematical Physics, Vol. 41, 2000, pp. 2968-2979. http://dx.doi.org/10.1063/1.533284
[24] E. G. Reyes, “On Geometrically Integrable Equations and Hierarchies of Pseudo-Spherical Type,” Contemporary Mathematics, Vol. 285, 2001, pp. 145-156.
[25] W. T. Wu, “Polynomial Equations-Solving and Its Applications,” Algorithms and Computation, Beijing, 1994, pp. 1-9.
[26] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, “The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems,” Studies in Applied Mathematics, Vol. 53, 1974, pp. 249-257.
[27] S. S. Chern and K. Tenenblat, “Pseudospherical Surfaces and Evolution Equations,” Studies in Applied Mathematics, Vol. 74, 1986, pp. 55-83.
[28] K. Konno and M. Wadati, “Simple Derivation of Backlund Transformation from Riccati Form of Inverse Method,” Progress of Theoretical Physics, Vol. 53, 1975, pp. 1652-1656. http://dx.doi.org/10.1143/PTP.53.1652
[29] R. Sasaki, “Soliton Equations and Pseudospherical Surfaces,” Nuclear Physics B, Vol. 154, 1979, pp. 343-357.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.