Journal of Applied Mathematics

The Simplest Equation Method and Its Application for Solving the Nonlinear NLSE, KGZ, GDS, DS, and GZ Equations

Yun-Mei Zhao, Ying-Hui He, and Yao Long

Full-text: Open access

Abstract

A good idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the elliptic-like equations are derived using the simplest equation method and the modified simplest equation method, and then the exact solutions of a class of nonlinear evolution equations which can be converted to the elliptic-like equation using travelling wave reduction are obtained. For example, the perturbed nonlinear Schrödinger’s equation (NLSE), the Klein-Gordon-Zakharov (KGZ) system, the generalized Davey-Stewartson (GDS) equations, the Davey-Stewartson (DS) equations, and the generalized Zakharov (GZ) equations are investigated and the exact solutions are presented using this method.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 960798, 7 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808261

Digital Object Identifier
doi:10.1155/2013/960798

Mathematical Reviews number (MathSciNet)
MR3133973

Zentralblatt MATH identifier
06950961

Citation

Zhao, Yun-Mei; He, Ying-Hui; Long, Yao. The Simplest Equation Method and Its Application for Solving the Nonlinear NLSE, KGZ, GDS, DS, and GZ Equations. J. Appl. Math. 2013 (2013), Article ID 960798, 7 pages. doi:10.1155/2013/960798. https://projecteuclid.org/euclid.jam/1394808261


Export citation

References

  • M. L. Wang, “Exact solutions for a compound KdV-Burgers equation,” Physics Letters A, vol. 213, no. 5-6, pp. 279–287, 1996.
  • M. L. Wang, Y. B. Zhou, and Z. B. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters A, vol. 216, no. 1–5, pp. 67–75, 1996.
  • Sirendaoreji and S. Jiong, “Auxiliary equation method for solving nonlinear partial differential equations,” Physics Letters A, vol. 309, no. 5-6, pp. 387–396, 2003.
  • Z. Y. Yan and H. Q. Zhang, “New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics,” Physics Letters A, vol. 252, no. 6, pp. 291–296, 1999.
  • E. Fan and J. Zhang, “Applications of the Jacobi elliptic function method to special-type nonlinear equations,” Physics Letters A, vol. 305, no. 6, pp. 383–392, 2002.
  • X.-H. Wu and J.-H. He, “EXP-function method and its application to nonlinear equations,” Chaos, Solitons & Fractals, vol. 38, no. 3, pp. 903–910, 2008.
  • H. A. Abdusalam, “On an improved complex tanh-function method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 99–106, 2005.
  • A. A. Soliman, “Extended improved tanh-function method for solving the nonlinear physical problems,” Acta Applicandae Mathematicae, vol. 104, no. 3, pp. 367–383, 2008.
  • S. B. Leble and N. V. Ustinov, “Darboux transforms, deep reductions and solitons,” Journal of Physics A, vol. 26, no. 19, pp. 5007–5016, 1993.
  • H.-C. Hu, X.-Y. Tang, S.-Y. Lou, and Q.-P. Liu, “Variable separation solutions obtained from Darboux transformations for the asymmetric Nizhnik-Novikov-Veselov system,” Chaos, Solitons & Fractals, vol. 22, no. 2, pp. 327–334, 2004.
  • M. L. Wang, X. Z. Li, and J. L. 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.
  • S. M. Guo and Y. B. Zhou, “The extended $(G'/G)$-expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations,” Applied Mathematics and Computation, vol. 215, no. 9, pp. 3214–3221, 2010.
  • N. A. Kudryashov, “Exact solitary waves of the Fisher equation,” Physics Letters A, vol. 342, no. 1-2, pp. 99–106, 2005.
  • N. A. Kudryashov, “Simplest equation method to look for exact solutions of nonlinear differential equations,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1217–1231, 2005.
  • N. K. Vitanov and Z. I. Dimitrova, “Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 10, pp. 2836–2845, 2010.
  • E. M. E. Zayed and S. A. Hoda Ibrahim, “Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method,” Chinese Physics Letters, vol. 29, no. 6, Article ID 060201, 2012.
  • A. J. Mohamad Jawad, M. D. Petković, and A. Biswas, “Modified simple equation method for nonlinear evolution equations,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 869–877, 2010.
  • A. Yildirim, A. Samiei Paghaleh, M. Mirzazadeh, H. Moosaei, and A. Biswas, “New exact traveling wave solutions for DS-I and DS-II equations,” Nonlinear Analysis: Modelling and Control, vol. 17, no. 3, pp. 369–378, 2012.
  • N. Taghizadeh, M. Mirzazadeh, A. Samiei Paghaleh, and J. Vahidi, “Exact solutions of nonlinear evolution equations by using the modified simple equation method,” Ain Shams Engineering Journal, vol. 3, no. 3, pp. 321–325, 2012.
  • Z.-Y. Zhang, Z.-H. Liu, X.-J. Miao, and Y.-Z. Chen, “New exact solutions to the perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 3064–3072, 2010.
  • Q. H. Shi, Q. Xiao, and X. J. Liu, “Extended wave solutions for a nonlinear Klein-Gordon-Zakharov system,” Applied Mathematics and Computation, vol. 218, no. 19, pp. 9922–9929, 2012.
  • Y. B. Zhou, M. L. Wang, and T. D. Miao, “The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations,” Physics Letters A, vol. 323, no. 1-2, pp. 77–88, 2004.
  • A. Davey and K. Stewartson, “On three-dimensional packets of surface waves,” Proceedings of the Royal Society A, vol. 338, pp. 101–110, 1974.
  • B. Malomed, D. Anderson, M. Lisak, M. L. Quiroga-Teixeiro, and L. Stenflo, “Dynamics of solitary waves in the Zakharov model equations,” Physical Review E, vol. 55, no. 1, pp. 962–968, 1997.