Lax Pair, Improved \(\varGamma \)-Riccati Bäcklund Transformation and Soliton-Like Solutions to Variable-Coefficient Higher-Order Nonlinear Schrödinger Equation in Optical Fibers

  • Yinglin Lu
  • Guangmei WeiEmail author
  • Xin Liu


In this paper, the higher-order nonlinear Schrödinger equation with variable-coefficient for the propagation of femtosecond pulses in optical fibers is analytically investigated. Lax pair is constructed via the Ablowitz-Kaup-Newell-Segur system, an infinite number of conservation laws are derived based on the Lax pair. Introducing an auxiliary function, an improved \(\varGamma \)-Riccati Bäcklund transformation is presented, which can generate successively nonlinear superposition formula. Moreover, one and two soliton-like solutions are obtained.


Variable-coefficient higher-order nonlinear Schrödinger equation Optical fiber Conservation law Improved \(\varGamma \)-Riccati Bäcklund transformation Soliton-like solution 



We would like to thank the Editor and Reviewers for their timely and valuable comments. We also express sincere thanks to J.Y. Wang and W.X. Zheng for their valuable discussions and helpful advices. This work has been supported by the National Natural Science Foundation of China under Grant No. 61471406.


  1. 1.
    Lamb, G.L.: Elements of Soliton Theory. Pure & Applied Mathematics. Wiley, New York (1980) zbMATHGoogle Scholar
  2. 2.
    Kanna, T., Tsoy, E.N., Akhmediev, N.: On the solution of multicomponent nonlinear Schrödinger equations. Phys. Lett. A 330, 224–229 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Xu, T., Zhang, C.Y., Wei, G.M., Li, J., Meng, X.H., Tian, B.: Symbolic-computation construction of transformations for a more generalized nonlinear Schrödinger equation with applications in inhomogeneous plasmas, optical fibers, viscous fluids and Bose-Einstein condensates. Eur. Phys. J. B 55, 323–332 (2007) CrossRefGoogle Scholar
  4. 4.
    Mollenauer, L.F., Stolen, R.H., Gordon, J.P.: Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys. Rev. Lett. 45, 1095–1098 (1980) CrossRefGoogle Scholar
  5. 5.
    Kodama, Y.J.: Optical solitons in a monomode fiber. J. Stat. Phys. 39, 597–614 (1985) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kodama, Y., Hasegawa, A.: Nonlinear pulse propagation in a monomode dielectric guide. IEEE J. Quantum Elect. 23, 510–524 (1987) CrossRefGoogle Scholar
  7. 7.
    Agrawal, G.P.: Effect of intrapulse stimulated Raman scattering on soliton-effect pulse compression in optical fibers. Opt. Lett. 15, 224–226 (1990) CrossRefGoogle Scholar
  8. 8.
    Li, Z., Li, L., Tian, H., Zhou, G.: New types of solitary wave solutions for the higher order nonlinear Schrödinger equation. Phys. Rev. Lett. 84, 4096 (2000) CrossRefGoogle Scholar
  9. 9.
    Yang, R.C., Li, L., Hao, R.Y., Li, Z.H., Zhou, G.S.: Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrödinger equation. Phys. Rev. E 71, 036616 (2005) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Abdullaeev, F.: Theory of Solitons in Inhomogeneous Media. Wiley, New York (1994) Google Scholar
  11. 11.
    Mahalingam, A., Alagesan, T.: Singularity structure analysis of inhomogeneous Hirota and higher order nonlinear Schrödinger equations. Chaos, Solitons & Fractals 25, 319–323 (2005) CrossRefzbMATHGoogle Scholar
  12. 12.
    Meng, X.H., Liu, W.J., Zhu, H.W., Zhang, C.Y., Tian, B.: Multi-soliton solutions and a Bäcklund transformation for a generalized variable-coefficient higher-order nonlinear Schrödinger equation with symbolic computation. Physica A 387, 97–107 (2008) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Yang, R.C., Hao, R.Y., Li, L., Li, Z.H., Zhou, G.S.: Dark soliton solution for higher-order nonlinear Schrödinger equation with variable coefficients. Opt. Commun. 242, 285–293 (2004) CrossRefGoogle Scholar
  14. 14.
    Li, L.X., Wang, M.L.: The (G′/G)-expansion method and travelling wave solutions for a higher-order nonlinear Schrödinger equation. Appl. Math. Comput. 208, 440–445 (2009) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Li, J., Zhang, H.Q.: Symbolic computation on the Darboux transformation for a generalized variable-coefficient higher-order nonlinear Schrödinger equation from fiber optics. J. Math. Anal. Appl. 365, 517–524 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ma, W.X., Strampp, W.: An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems. Phys. Lett. A 185, 277 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sanukia, H., Konnoa, K.: Conservation laws of sine-Gordon equation. Phys. Lett. A 48, 221–222 (1974) CrossRefGoogle Scholar
  18. 18.
    Konno, K., Sanuki, H., Ichikawa, Y.H.: Conservation laws of nonlinear-evolution equations. Prog. Theor. Phys. 52, 886–889 (1974) CrossRefGoogle Scholar
  19. 19.
    Wadati, M., Konno, K., Ichikawa, Y.: A generalization of inverse scattering method. J. Phys. Soc. Jpn. 46, 1965–1966 (1979) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rao, J.A., Rangwala, A.A.: In: Lakshmanan, M. (ed.) Soliton: Introduction and Applications. Springer, Berlin (1988) Google Scholar
  21. 21.
    Liu, Y.P., Gao, Y.T., Wei, G.M.: An improved \(\Gamma\)-Riccati Bäcklund transformation and its applications for the inhomogeneous nonlinear Schrödinger model from plasma physics and nonlinear optics. Physica A 391, 535–543 (2012) CrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.LMIB and School of Mathematics and System SciencesBeihang UniversityBeijingChina
  2. 2.The PLA Information Engineering UniversityZhengzhouChina

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