Abstract and Applied Analysis

Exact Solutions and Conservation Laws of the Drinfel’d-Sokolov-Wilson System

Catherine Matjila, Ben Muatjetjeja, and Chaudry Masood Khalique

Full-text: Open access

Abstract

We study the Drinfel'd-Sokolov-Wilson system, which was introduced as a model of water waves. Firstly we obtain exact solutions of this system using the (G/G)-expansion method. In addition to exact solutions we also construct conservation laws for the underlying system using Noether's approach.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 271960, 6 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412278615

Digital Object Identifier
doi:10.1155/2014/271960

Mathematical Reviews number (MathSciNet)
MR3191028

Zentralblatt MATH identifier
07022062

Citation

Matjila, Catherine; Muatjetjeja, Ben; Khalique, Chaudry Masood. Exact Solutions and Conservation Laws of the Drinfel’d-Sokolov-Wilson System. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 271960, 6 pages. doi:10.1155/2014/271960. https://projecteuclid.org/euclid.aaa/1412278615


Export citation

References

  • Z. Wen, Z. Liu, and M. Song, “New exact solutions for the classical Drinfel'd-Sokolov-Wilson equation,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2349–2358, 2009.
  • R.-x. Yao and Z.-b. Li, “New exact solutions for three nonlinear evolution equations,” Physics Letters A, vol. 297, no. 3-4, pp. 196–204, 2002.
  • C. Liu and X. Liu, “Exact solutions of the classical Drinfel'd-Sokolov-Wilson equations and the relations among the solutions,” Physics Letters A, vol. 303, no. 2-3, pp. 197–203, 2002.
  • R. Hirota, B. Grammaticos, and A. Ramani, “Soliton structure of the Drinfel'd-Sokolov-Wilson equation,” Journal of Mathematical Physics, vol. 27, no. 6, pp. 1499–1505, 1986.
  • E. Fan, “An algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinear evolution equations,” Journal of Physics A: Mathematical and General, vol. 36, no. 25, pp. 7009–7026, 2003.
  • Y. Yao, “Abundant families of new traveling wave solutions for the coupled Drinfel'd-Sokolov-Wilson equation,” Chaos, Solitons & Fractals, vol. 24, no. 1, pp. 301–307, 2005.
  • M. Inc, “On numerical doubly periodic wave solutions of the coupled Drinfel'd-Sokolov-Wilson equation by the decomposition method,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 421–430, 2006.
  • X.-Q. Zhao and H.-Y. Zhi, “An improved $F$-expansion method and its application to coupled Drinfel'd-Sokolov-Wilson equation,” Communications in Theoretical Physics, vol. 50, no. 2, pp. 309–314, 2008.
  • 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.
  • Z. Y. Yan and H. Q. Zhang, “On a new algorithm of constructing solitary wave solutions for systems of nonlinear evolution equations in mathematical physics,” Applied Mathematics and Mechanics, vol. 21, no. 4, pp. 383–388, 2000.
  • M. L. Wang, “Solitary wave solutions for variant Boussinesq equations,” Physics Letters A, vol. 199, no. 3-4, pp. 169–172, 1995.
  • A. M. Wazwaz, “Compactons and solitary patterns structures for variants of the KdV and the KP equations,” Applied Mathematics and Computation, vol. 139, no. 1, pp. 37–54, 2003.
  • E. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212–218, 2000.
  • A. M. Wazwaz, “The tanh method for compact and noncompact solutions for variants of the KdV-Burger and the $K(n,n)$-Burger equations,” Physica D, vol. 213, no. 2, pp. 147–151, 2006.
  • M. Wang, X. Li, and J. Zhang, “The $({G}^{'}/G)$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008.
  • P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Communications on Pure and Applied Mathematics, vol. 21, pp. 467–490, 1968.
  • T. B. Benjamin, “The stability of solitary waves,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 328, pp. 153–183, 1972.
  • R. J. Knops and C. A. Stuart, “Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity,” Archive for Rational Mechanics and Analysis, vol. 86, no. 3, pp. 233–249, 1984.
  • R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, Basel, Switzerland, 1992.
  • E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, Berlin, Germany, 1996.
  • A. Sjöberg, “Double reduction of PDEs from the association of symmetries with conservation laws with applications,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 608–616, 2007.
  • A. H. Bokhari, A. Y. Al-Dweik, A. H. Kara, F. M. Mahomed, and F. D. Zaman, “Double reduction of a nonlinear $(2+1)$ wave equation via conservation laws,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1244–1253, 2011.
  • G. L. Caraffini and M. Galvani, “Symmetries and exact solutions via conservation laws for some partial differential equations of mathematical physics,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1474–1484, 2012.
  • E. Noether, “Invariante variationsprobleme,” Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu G öttingen, Mathematisch-Physikalische Klasse, Heft, vol. 2, pp. 235–257, 1918.
  • P. S. Laplace, Traite de Mecanique Celeste, vol. 1, Paris, France, 1978, (English translation Celestial Mechanics, New York, NY, USA, 1966).
  • H. Steudel, “Über die Zuordnung zwischen Invarianzeigenschaften und Erhaltungssätzen,” Zeitschrift für Naturforschung, vol. 17a, pp. 129–132, 1962.
  • P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, NY, USA, 1993.
  • S. C. Anco and G. Bluman, “Direct construction method for conservation laws of partial differential equations. I. Examples of conservation law classifications,” European Journal of Applied Mathematics, vol. 13, no. 5, pp. 545–566, 2002.
  • A. H. Kara and F. M. Mahomed, “Relationship between symmetries and conservation laws,” International Journal of Theoretical Physics, vol. 39, no. 1, pp. 23–40, 2000.
  • A. H. Kara and F. M. Mahomed, “Noether-type symmetries and conservation laws via partial Lagrangians,” Nonlinear Dynamics, vol. 45, no. 3-4, pp. 367–383, 2006.
  • R. Naz, F. M. Mahomed, and T. Hayat, “Conservation laws for third-order variant Boussinesq system,” Applied Mathematics Letters, vol. 23, no. 8, pp. 883–886, 2010. \endinput