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Science in China Series A: Mathematics

, Volume 50, Issue 6, pp 773–785 | Cite as

Dynamical understanding of loop soliton solution for several nonlinear wave equations

  • Ji-bin Li
Article

Abstract

It has been found that some nonlinear wave equations have one-loop soliton solutions. What is the dynamical behavior of the so-called one-loop soliton solution? To answer this question, the travelling wave solutions for four nonlinear wave equations are discussed. Exact explicit parametric representations of some special travelling wave solutions are given. The results of this paper show that a loop solution consists of three different breaking travelling wave solutions. It is not one real loop soliton travelling wave solution.

Keywords

planar dynamical system homoclinic orbit solitary wave solution one-loop soliton solution periodic wave solution bifurcation nonlinear wave equation 

MSC(2000)

34C37 34C23 74J30 58Z05 

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References

  1. [1]
    Vakhnenko V O. High-frequency soliton-like waves in a relaxing medium. J Math Phys, 40: 2011–2020 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Vakhnenko V O, Parkes E J. The two loop soliton solution of the Vakhnenko equation. Nonlinearity, 11: 1457–1464 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Morrison T P, Parkes E J, Vakhnenko V O. The N-loop soliton solution of the Vakhnenko equation. Nonlinearity, 12: 1427–1437 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Morrison T P, Parkes E J. The N-loop soliton solution of the modified Vakhnenko equation (a new nonlinear evolution equation). Chaos, Solitons and Fractals, 16: 13–26 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Sakovich A, Sakovich S. Solitary wave solutions of the short pulse equation. J Phys A Math Gen, 39: L361–367 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Schafer T, Wayne C E. Propagation of ultra-short opical pulses in cubic nonlinear media. Physica D, 196: 90–105 (2004)CrossRefMathSciNetGoogle Scholar
  7. [7]
    Tzirtzilakis E, Marinakis V, Apokis C, Bountis T. Soliton-like solutions of higher order wave equations of the Korteweg-de-Vries Type. J Math Phys, 43: 6151–6161 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Tzirtzilakis E, Xenos M, Marinakis V, Bountis T. Interactions and stability of solitary waves in shallow water. Chaos, Solitons and Fractals, 14: 87–95 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Fokas A S. On class of physically important integrable equations. Physica D, 87: 145–150 (1995)CrossRefMathSciNetGoogle Scholar
  10. [10]
    Li J B, Wu J H, Zhu H P. Travelling waves for an Integrable Higher Order KdV Type Wave Equations. International Journal of Bifurcation and Chaos, 16(8): 2235–2260 (2006)CrossRefMathSciNetGoogle Scholar
  11. [11]
    Li J B, Pai H H. On the Study of Sigular Nonlinear Travelling Wave Equations: Dynamical Syotem Appwach. Beijing: Science Press, 2007Google Scholar
  12. [12]
    Byrd P F, Fridman M D. Handbook of Elliptic Integrals for Engineers and Sciensists. Berlin: Springer, 1971Google Scholar

Copyright information

© Science in China Press 2007

Authors and Affiliations

  • Ji-bin Li
    • 1
    • 2
  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaChina
  2. 2.Kunming University of Science and TechnologyKunmingChina

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