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Acta Mathematica Sinica, English Series

, Volume 28, Issue 5, pp 969–974 | Cite as

Bäcklund transformation and conservation laws for the variable-coefficient N-coupled nonlinear Schrödinger equations with symbolic computation

  • Xiang Hua MengEmail author
  • Bo Tian
  • Tao Xu
  • Hai Qiang Zhang
Article

Abstract

Considering the integrable properties for the coupled equations, the variable-coefficient Ncoupled nonlinear Schrödinger equations are under investigation analytically in this paper. Based on the Lax pair with the nonisospectral parameter, a Bäcklund transformation for such a coupled system denoting in the Γ functions is constructed with the one-solitonic solution given as the application sample. Furthermore, an infinite number of conservation laws are obtained using symbolic computation.

Keywords

Variable-coefficient N-coupled nonlinear Schrödinger equations Bäcklund transformation conservation laws solitonic solution symbolic computation 

MR(2000) Subject Classification

35A20 

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Supplementary material

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References

  1. [1]
    Agrawal, G. P.: Nonlinear Fiber Optics, Academic Press, New York, 1995Google Scholar
  2. [2]
    Ablowitz, M. J., Clarkson, P. A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991zbMATHCrossRefGoogle Scholar
  3. [3]
    Shu, J., Zhang, J.: Sharp criterion of global existence for nonlinear Schrödinger equation with a harmonic potential. Acta Mathematica Sinica, English Series, 25, 537–544 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Yuan, Y. R.: The modified characteristic finite difference fractional steps method for nonlinear coupled system of multilayer fluid dynamics in porous media. Acta Mathematica Scientia, 29, 858–872 (2009)zbMATHGoogle Scholar
  5. [5]
    Sun, Y. P., Tam, H.W.: Grammian solutions and pfaffianization of a non-isospectral and variable-coefficient Kadomtsev-Petviashvili equation. J. Math. Anal. Appl., 343, 810–817 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Brugarino, T., Sciacca, M.: Singularity analysis and integrability for a HNLS equation governing pulse propagation in a generic fiber optics. Opt. Commun., 262, 250–256 (2006)CrossRefGoogle Scholar
  7. [7]
    Xu, T., Li, J., Zhang, H. Q., et al.: Integrable aspects and applications of a generalized inhomogeneous N-coupled nonlinear Schrödinger system in plasmas and optical fibers via symbolic computation. Phys. Lett. A, 372, 1990–2001 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Meng, X. H., Tian, B., Xu, T., et al.: Solitonic solutions and Bäcklund transformation for the inhomogeneous N-coupled nonlinear Schrödinger equations. Physica A, 388(2–3), 209–217 (2009)MathSciNetCrossRefGoogle Scholar
  9. [9]
    Nakkeeran, K.: Exact soliton solutions for a family of N coupled nonlinear Schrödinger equations in optical fiber media. Phys. Rev. E, 62, 1313–1321 (2000)MathSciNetCrossRefGoogle Scholar
  10. [10]
    Wu, L., Yang, Q., Zhang, J. F.: Bright solitons on a continuous wave background for the inhomogeneous nonlinear Schrödinger equation in plasma. J. Phys. A, 39, 11947–11953 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Khawaja, U. A.: Lax pairs of time-dependent Gross-Pitaevskii equation. J. Phys. A, 39, 9679–9691 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Lax, P. D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math., 21, 467–490 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Ablowitz, M. J., Kaup, D. J., Newell, A. C., et al.: Method for solving Sine-Gordon equation. Phys. Rev. Lett., 30, 1262–1264 (1973)MathSciNetCrossRefGoogle Scholar
  14. [14]
    Wadati, M., Sanuki, H., Konno, K.: Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws. Prog. Theor. Phys., 53(2), 419–436 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Lamb, G. L., Jr.: Element of Soliton Theory, John Wiley & Sons, New York, 1980Google Scholar
  16. [16]
    Tian, B., Gao, Y. T.: Symbolic-computation study of the perturbed nonlinear Schrödinger model in inhomogeneous optical fibers. Phys. Lett. A, 342(3), 228–236 (2005)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Xiang Hua Meng
    • 1
    • 2
    Email author
  • Bo Tian
    • 1
  • Tao Xu
    • 1
  • Hai Qiang Zhang
    • 1
  1. 1.School of ScienceBeijing University of Posts and TelecommunicationsBeijingP. R. China
  2. 2.Department of Mathematics, School of Applied ScienceBeijing Information Science and Technology UniversityBeijingP. R. China

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