Nonlinear Dynamics

, Volume 88, Issue 3, pp 1615–1622 | Cite as

Modulation instability, conservation laws and soliton solutions for an inhomogeneous discrete nonlinear Schrödinger equation

  • Hui-Qin Hao
  • Rui Guo
  • Jian-Wen Zhang
Original Paper


In this paper, an inhomogeneous discrete nonlinear Schrödinger equation is analytically investigated. The modulation instability condition and conservation laws are derived. By virtue of the discrete Darboux transformation, two types of explicit solutions on the vanishing and non-vanishing backgrounds are generated. Those results might be useful in the study of solitons propagation in discrete optical fibers.


An inhomogeneous discrete nonlinear Schrödinger equation Soliton Modulation instability Conservation laws Discrete Darboux transformation 



We express our sincere thanks to each member of our discussion group for their suggestions. This work has been supported by the Special Funds of the National Natural Science Foundation of China under Grant No. 11347165 and 61405137, by the Shanxi Province Science Foundation for Youths under Grant Nos. 2015021008 and 2014011005-4.


  1. 1.
    Hasegawa, A., Kodama, Y.: Solitons in optical communications. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
  2. 2.
    Ablowitz, M.J.: Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  3. 3.
    Agrawal, G.P.: Nonlinear fiber optics. Academic Press, San Diego (2001)zbMATHGoogle Scholar
  4. 4.
    Biswas, A., Khalique, C.M.: Stationary solutions for nonlinear dispersive Schrödinger’s equation. Nonlinear Dyn. 63, 623–626 (2011)CrossRefGoogle Scholar
  5. 5.
    Kohl, R., Biswas, A., Milovic, D., Zerrad, E.: Optical soliton perturbation in a non-Kerr law media. Opt. Laser Tech. 40, 647–655 (2008)CrossRefGoogle Scholar
  6. 6.
    Ghodrat, E., Anjan, B.: The G’/G method and 1-soliton solution of Davey–Stewartson equation. Math. Comp. Model. 53(5–6), 694–698 (2011)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Ghodrat, E., Krishnan, E.V., Manel, L., Essaid, Z., Anjan, B.: Analytical and numerical solutions for Davey–Stewartson equation with power lawnonlinearity. Waves Rand. Comp. Med. 21(4), 559–590 (2011)CrossRefzbMATHGoogle Scholar
  8. 8.
    Hossein, J., Atefe, S., Yahya, T., Anjan, B.: The first integral method and traveling wave solutions to Davey–Stewartson equation. Nonlinear Anal. Model. Contr. 17(2), 182–193 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Mirzazadeh, M., Eslami, M., Zerrad, E., Mahmood, M.F., Biswas, A., Belic, M.: Optical solitons in nonlinear directional couplers by sine–cosine function method and Bernoulli’s equation approach. Nonlinear Dyn. 81, 1933–1949 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mirzazadeh, M., Eslami, M., Savescu, M., Bhrawy, A.H., Alshaery, A.A., Hilal, E.M., Biswas, A.: Optical solitons in DWDM system with spatio-temporal dispersion. J. Nonlinear Opt. Phys. Mater. 24, 1550006 (2015)CrossRefGoogle Scholar
  11. 11.
    Zhou, Q., Zhu, Q.P., Savescu, M., Bhrawy, A., Biswas, A.: Optical solitons with nonlinear dispersion in parabolic law medium. Proc. Romanian Acad. Ser. A 16, 152–159 (2015)MathSciNetGoogle Scholar
  12. 12.
    Zhou, Q., Zhu, Q.P., Liu, Y.X., Yu, H., Wei, C., Yao, P., Bhrawy, A., Biswas, A.: Bright, dark and singular optical solitons in cascaded system. Laser Phys. 25, 025402 (2015)CrossRefGoogle Scholar
  13. 13.
    Zhou, Q., Yao, D., Chen, F.: Analytical study of optical solitons in media with Kerr and parabolic law nonlinearities. J. Mod. Opt. 60, 1652C1657 (2013)Google Scholar
  14. 14.
    Zhou, Q., Yao, D., Chen, F., Li, W.: Optical solitons in gasfilled, hollow-core photonic crystal fibers with inter-modal dispersion and self-steepening. J. Mod. Opt. 60, 854–859 (2013)CrossRefzbMATHGoogle Scholar
  15. 15.
    Mirzazadeh, M., Arnous, A.H., Mahmood, M.F., Zerrad, E., Biswas, A.: Soliton solutions to resonant nonlinear Schrödinger’s equation with time-dependent coeffcients by trial solution approach. Nonlinear Dyn. 81, 277–282 (2015)CrossRefzbMATHGoogle Scholar
  16. 16.
    Tu, J.M., Tian, S.F., Xu, M.J., Song, X.Q., Zhang, T.T.: Bäcklund transformation, infinite conservation laws and periodic wave solutions of a generalized (3+1)-dimensional nonlinear wave in liquid with gas bubbles. Nonlinear Dyn. 83, 1199–1215 (2016)CrossRefzbMATHGoogle Scholar
  17. 17.
    Tian, S.F., Zhou, S.W., Jiang, W.Y., Zhang, H.Q.: Analytic solutions, Darboux transformation operators and supersymmetry for a generalized one-dimensional time-dependent Schrödinger equation. Appl. Math. Comput. 218, 7308–7321 (2012)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Liu, W.J., Pan, N., Huang, L.G., Lei, M.: Soliton interactions for coupled nonlinear Schrödinger equations with symbolic computation. Nonlinear Dyn. 78, 755–770 (2014)CrossRefGoogle Scholar
  19. 19.
    Liu, W.J., Huang, L.G., Li, Y.Q., Pan, N., Lei, M.: Interactions of dromion-like structures in the (1+1) dimension variable coefficient nonlinear Schrödinger equation. Appl. Math. Lett. 39, 91–95 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Qi, F.H., Xu, X.G., Wang, P.: Rogue wave solutions for the coupled cubic-quintic nonlinear Schrödinger equations with variable coefficients. Appl. Math. Lett. 54, 60–65 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Qi, F.H., Ju, H.M., Meng, X.H., Li, J.: Conservation laws and Darboux transformation for the coupled cubic–quintic nonlinear Schrodinger equations with variable coefficients in nonlinear optics. Nonlinear Dyn. 77, 1331–1337 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhao, H.H., Zhao, X.J., Hao, Q.: Breather-to-soliton conversions and nonlinear wave interactions in a coupled Hirota system. Appl. Math. Lett. 61, 8–12 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Guo, R., Liu, Y.F., Hao, H.Q., Qi, F.H.: Coherently coupled solitons, breathers and rogue waves for polarized optical waves in an isotropic medium. Nonlinear Dyn. 80, 1221–1230 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Guo, R., Zhao, H.H., Wang, Y.: A higher-order coupled nonlinear Schrödinger system: solitons, breathers, and rogue wave solutions. Nonlinear Dyn. 83, 2475–2484 (2016)CrossRefzbMATHGoogle Scholar
  25. 25.
    Li, L., Li, Z.H., Li, S.Q., Zhou, G.S.: Modulation instability and solitons on a cw background in inhomogeneous optical fiber media. Opt. Commun. 234, 169–176 (2004)CrossRefGoogle Scholar
  26. 26.
    Kevrekidis, P.G.: The discrete nonlinear Schrödinger equation: mathematical analysis. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
  27. 27.
    Toda, M.: Theory of nonlinear lattices. Springer, Berlin (1981)CrossRefzbMATHGoogle Scholar
  28. 28.
    Konotop, V.V., Chubykalo, O.A., Vázquez, L.: Dynamics and interaction of solitons on an integrable inhomogeneous lattice. Phys. Rev. E 48, 563–568 (1993)CrossRefGoogle Scholar
  29. 29.
    Scharf, R., Bishop, A.R.: Properties of the nonlinear Schrödinger equation on a lattice. Phys. Rev. A 43, 6535–6544 (1991)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Ankiewicz, A., Akhmediev, N., Soto-Crespo, J.M.: Discrete rogue waves of the Ablowitz-Ladik and Hirota equations. Phys. Rev. E 82, 026602 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Qin, Z.Y.: A generalized AblowitzCLadik hierarchy, multi-Hamiltonian structure and Darboux transformation. J. Math. Phys. 49, 063505 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Mohamadou, A., Lantchio Tiofack, C.G., Kofané, T.C.: Stability analysis of plane wave solutions of the generalized AblowitzCLadik system. Phys. Scr. 74, 718–725 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kamchatnov, A.M.: Nonlinear periodic waves and their modulations. World Science Press, Hong Kong (2000)Google Scholar
  34. 34.
    Wright, O.C.: Homoclinic connections of unstable plane waves of the long wave short wave equations. Stud. Appl. Math. 117, 71–93 (2006)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Murali, R., Porsezian, K., Kofané, T.C., Mohamadou, A.: Modulational instability and exact solutions of the discrete cubicCquintic GinzburgCLandau equation. J. Phys. A Math. Theor. 43, 165001 (2010)CrossRefzbMATHGoogle Scholar
  36. 36.
    Ji, J., Zhang, D.J., Zhang, J.J.: Conservation laws of a discrete soliton. Commun. Theory Phys. 49, 1105–1108 (2008)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Abdul, H.K., Houria, T., Anjan, B.: Conservation laws of the Bretherton equation. Appl. Math. Inf. Sci. 7(3), 877–879 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.School of MathematicsTaiyuan University of TechnologyTaiyuanChina

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