Abstract and Applied Analysis

New Exact Solitary Wave Solutions of a Coupled Nonlinear Wave Equation

XiaoHua Liu and CaiXia He

Full-text: Open access

Abstract

By using the theory of planar dynamical systems to a coupled nonlinear wave equation, the existence of bell-shaped solitary wave solutions, kink-shaped solitary wave solutions, and periodic wave solutions is obtained. Under the different parametric values, various sufficient conditions to guarantee the existence of the above solutions are given. With the help of three different undetermined coefficient methods, we investigated the new exact explicit expression of all three bell-shaped solitary wave solutions and one kink solitary wave solutions with nonzero asymptotic value for a coupled nonlinear wave equation. The solutions cannot be deduced from the former references.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 301645, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393512095

Digital Object Identifier
doi:10.1155/2013/301645

Mathematical Reviews number (MathSciNet)
MR3121505

Zentralblatt MATH identifier
1295.35176

Citation

Liu, XiaoHua; He, CaiXia. New Exact Solitary Wave Solutions of a Coupled Nonlinear Wave Equation. Abstr. Appl. Anal. 2013 (2013), Article ID 301645, 7 pages. doi:10.1155/2013/301645. https://projecteuclid.org/euclid.aaa/1393512095


Export citation

References

  • M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, UK, 1991.
  • R. Hirota, “Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971.
  • J.-H. He, “Variational iteration method–-some recent results and new interpretations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 3–17, 2007.
  • 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, pp. 67–75, 1996.
  • G. L. Lamb, “Bäcklund transformations for certain nonlinear evolution equations,” Journal of Mathematical Physics, vol. 15, no. 12, pp. 2157–2165, 1974.
  • A. M. Wazawaz, “New traveling wave solutions of differential physical structures to generalized BBM equation,” Physics Letters A, vol. 355, no. 4-5, pp. 358–362, 2006.
  • E. V. Krishnan, “On the Itô-type coupled nonlinear wave equation,” Journal of the Physical Society of Japan, vol. 55, no. 11, pp. 3753–3755, 1986.
  • S. Zhang, “A generalized new auxiliary equation method and its application to the $(2+1)$-dimensional breaking soliton equations,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 510–516, 2007.
  • E. Yomba, “A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations,” Physics Letters A, vol. 372, no. 7, pp. 1048–1060, 2008.
  • F. Kangalgil and F. Ayaz, “New exact travelling wave solutions for the Ostrovsky equation,” Physics Letters A, vol. 372, no. 11, pp. 1831–1835, 2008.
  • 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.
  • X. Liu, W. Zhang, and Z. Li, “Application of improved $({G}^{'}/G)$-expansion method to traveling wave solutions of two nonlinear evolution equations,” Advances in Applied Mathematics and Mechanics, vol. 4, no. 1, pp. 122–130, 2012.
  • R. Hirota and J. Satsuma, “Soliton solutions of a coupled Korteweg-de Vries equation,” Physics Letters A, vol. 85, no. 8-9, pp. 407–408, 1981.
  • B. A. Kupershmidt, “A coupled Korteweg-de Vries equation with dispersion,” Journal of Physics A, vol. 18, no. 10, pp. L571–L573, 1985.
  • M. Ito, “Symmetries and conservation laws of a coupled nonlinear wave equation,” Physics Letters A, vol. 91, no. 7, pp. 335–338, 1982.
  • S. Kawamoto, “Cusp soliton solutions of the Itô-type coupled nonlinear wave equation,” Journal of the Physical Society of Japan, vol. 53, no. 4, pp. 1203–1205, 1984.
  • D. C. Lu and G. J. Yang, “Compacton solutions and peakon solutions for a coupled nonlinear wave equation,” International Journal of Nonlinear Science, vol. 4, no. 1, pp. 31–36, 2007.
  • C. Guha-Roy, B. Bagchi, and D. K. Sinha, “Traveling-wave solutions and the coupled Korteweg-de Vries equation,” Journal of Mathematical Physics, vol. 27, no. 10, pp. 2558–2560, 1986.
  • A. Cavaglia, A. Fring, and B. Bagchi, “$PT$-symmetry breaking in complex nonlinear wave equations and their deformations,” Journal of Physics A, vol. 44, no. 32, Article ID 325201, 2011.
  • C. Guha-Roy, “Solitary wave solutions of a system of coupled nonlinear equations,” Journal of Mathematical Physics, vol. 28, no. 9, pp. 2087–2088, 1987.
  • M. Wadati, “Wave propagation in nonlinear lattice I, II,” Journal of the Physical Society of Japan, vol. 38, pp. 673–686, 1975.
  • C. Guha-Roy, “Exact solutions to a coupled nonlinear equation,” International Journal of Theoretical Physics, vol. 27, no. 4, pp. 447–450, 1988.
  • B. Q. Lu, Z. L. Pan, B. Z. Qu, and X. F. Jiang, “Solitary wave solutions for some systems of coupled nonlinear equations,” Physics Letters A, vol. 180, no. 1-2, pp. 61–64, 1993.
  • S. N. Chow and J. K. Hale, Method of Bifurcation Theory, Springer, New York, NY, USA, 1981.
  • J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, NY, USA, 1983.
  • L. Perko, Differential Equations and Dynamical Systems, Springer, New York, NY, USA, 1991.