Nonlinear Dynamics

, Volume 83, Issue 1–2, pp 1043–1052 | Cite as

Exact \(\varvec{N}\)-soliton solutions and dynamics of a new AKNS equation with time-dependent coefficients

  • Sheng Zhang
  • Xudong Gao
Original Paper


Based on the corresponding bilinear forms, we obtain one-, two- and three-soliton solutions of a new AKNS equation with time-dependent coefficients by using Hirota’s bilinear method. From these obtained solutions, uniform formulae of exact N-soliton solutions of the AKNS equation are summarized. It is graphically shown that the dynamical evolutions of such soliton solutions with time-dependent functions of the AKNS equation possess time-varying speeds and amplitudes in the process of propagations.


Bilinear form Soliton solution  Hirota’s bilinear method AKNS equation with time-dependent coefficients 



We would like to express our sincerest thanks to the referee for the valuable suggestions and comments which lead to further improvement of our original manuscript. This work was supported by the Natural Science Foundation of Liaoning Province of China (L2012404), the Ph.D. Start-up Funds of Liaoning Province of China (20141137) and Bohai University (bsqd2013025), the Liaoning BaiQianWan Talents Program (2013921055) and the Natural Science Foundation of China (11371071).


  1. 1.
    Drazin, P.G., Johnson, R.S.: Solitons: An Introduction. Cambridge University Press, New York (1989)CrossRefMATHGoogle Scholar
  2. 2.
    Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg–de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967)CrossRefMATHGoogle Scholar
  3. 3.
    Miurs, M.R.: Bäcklund Transformation. Springer, Berlin (1978)Google Scholar
  4. 4.
    Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60, 650–654 (1992)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Wang, M.L.: Exact solutions for a compound KdV–Burgers equation. Phys. Lett. A 213, 279–287 (1996)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Liu, S.K., Fu, Z.T., Liu, S.D., Zhao, Q.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 289, 69–74 (2001)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Fan, E.G.: Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems. Phys. Lett. A 300, 243–249 (2002)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Fan, E.G.: Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics. Chaos Solitons Fractals 16, 819–839 (2003)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Yomba, E.: The modified extended Fan sub-equation method and its application to the (2+1)-dimensional Broer–Kaup–Kupershmidt equation. Chaos Solitons Fractals 27, 187–196 (2007)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Zhang, S., Xia, T.C.: A generalized auxiliary equation method and its application to (2+1)-dimensional asymmetric Nizhnik–Novikov–Vesselov equations. J. Phys. A Math. Theor. 40, 227–248 (2007)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30, 700–708 (2006)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Zhang, S.: Exact solutions of a KdV equation with variable coefficients via Exp-function method. Nonlin. Dyn. 52, 11–17 (2008)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Ma, W.X., Lee, J.H.: A transformed rational function method and exact solutions to 3+1 dimensional Jimbo–Miwa equation. Chaos Solitons Fractals 42, 1356–1363 (2009)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Dai, C.Q., Wang, Y.Y., Tian, Q., Zhang, J.F.: The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation. Ann. Phys. 327, 512–521 (2012)CrossRefMATHGoogle Scholar
  15. 15.
    Antonova, M., Biswas, A.: Adiabatic parameter dynamics of perturbed solitary waves. Commun. Nonlin. Sci. Numer. Simul. 14, 734–748 (2009)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Mirzazadeh, M., Eslami, M., Biswas, A.: 1-Soliton solution to KdV6 equation. Nonlin. Dyn. 80, 387–396 (2015)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Bhrawy, A.H., Abdelkawy, M.A., Biswas, A.: Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi’s elliptic function method. Commun. Nonlin. Sci. Numer. Simul. 18, 915–925 (2013)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Bhrawy, A.H., Biswas, A., Javidi, M., Ma, W.X., Pinar, Z., Yildirim, A.: New solutions for (1+1)-dimensional and (2+1)-dimensional Kaup–Kupershmidt equations. Results Math. 63, 675–686 (2013)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Bhrawy, A.H., Abdelkawy, M.A., Kumar, S., Biswas, A.: Solitons and other solutions to Kadomtsev–Petviashvili equation of b-type. Rom. J. Phys. 58, 729–748 (2013)MathSciNetGoogle Scholar
  20. 20.
    Ebadi, G., Mojaver, A., Triki, H., Yildirim, A., Biswas, A.: Topological solitons and other solutions to the (3+1)-dimensional extended Kadomtsev–Petviashvili equation with power law nonlinearity. Rom. J. Phys. 58, 3–14 (2013)MathSciNetGoogle Scholar
  21. 21.
    Triki, H., Kara, A.H., Bhrawy, A., Biswas, A.: Soliton solution and conservation law of Gear–Grimshaw model for shallow water waves. Acta Phys. Pol. A 125, 1099–1106 (2014)CrossRefGoogle Scholar
  22. 22.
    Triki, H., Mirzazadeh, M., Bhrawy, A.H., Razborova, P., Biswas, A.: Soliton and other solutions to long-wave short-wave interaction equation. Rom. J. Phys. 60, 72–86 (2015)Google Scholar
  23. 23.
    Razborova, P., Moraru, L., Biswas, A.: Perturbation of dispersive shallow water waves with Rosenau–KdV–RLW equation with power law nonlinearity. Rom. J. Phys. 59, 658–676 (2014)Google Scholar
  24. 24.
    Razborova, P., Triki, H., Biswas, A.: Perturbation of dispersive shallow water waves. Ocean Eng. 63, 1–7 (2013)CrossRefGoogle Scholar
  25. 25.
    Dai, C.Q., Wang, X.G., Zhou, G.Q.: Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials. Phys. Rev. A 89, 013834 (2014)CrossRefGoogle Scholar
  26. 26.
    Fan, E.G., Chow, K.W., Li, J.H.: On doubly periodic standing wave solutions of the coupled higgs field equation. Stud. Appl. Math. 128, 86–105 (2012)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Zayed, E.M.E., Abdelaziz, M.A.M.: An extended (G\(^{\prime }\)/G)- expansion method and its applications to the (2+1)-dimensional nonlinear evolution equations. WSEAS Trans. Math. 11, 1039–1047 (2012)Google Scholar
  28. 28.
    Wang, Y.L.: Variable-coefficient simplest equation method for solving nonlinear evolution equations in mathematical physics. WSEAS Trans. Math. 12, 512–520 (2013)Google Scholar
  29. 29.
    Kamenov, O.Y.: New periodic exact solutions of the Kuramoto–Sivashinsky evolution equation. WSEAS Trans. Math. 13, 345–352 (2014)Google Scholar
  30. 30.
    Hirota, R.: Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)CrossRefMATHGoogle Scholar
  31. 31.
    Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, New York (2004)CrossRefMATHGoogle Scholar
  32. 32.
    Balashov, M.V.: A property of the ansatz of Hirota’s method for quasilinear parabolic equations. Math. Notes 71, 339–354 (2002)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Zuo, D.W., Gao, Y.T., Meng, G.Q., Shen, Y.J., Yu, X.: Multi-soliton solutions for the three-coupled KdV equations engendered by the Neumann system. Nonlin. Dyn. 75, 701–708 (2014)CrossRefMathSciNetMATHGoogle Scholar
  34. 34.
    Mcarthur, I., Yung, C.M.: Hirota bilinear form for the super-KdV hierarchy. Mod. Phys. Lett. A 8, 1739–1745 (1993)CrossRefMathSciNetMATHGoogle Scholar
  35. 35.
    Liu, Q.P., Hu, X.B., Zhang, M.X.: Supersymmetric modified Korteweg–de Vries equation: bilinear approach. Nonlinearity 18, 1597–1603 (2005)CrossRefMathSciNetMATHGoogle Scholar
  36. 36.
    Chen, D.Y.: Introduction to Soliton. Science Press, Beijing (2006)Google Scholar
  37. 37.
    Wazwaz, A.M.: The Hirota’s bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada–Kotera–Kadomtsev–Petviashvili equation. Appl. Math. Comput. 200, 160–166 (2008)CrossRefMathSciNetMATHGoogle Scholar
  38. 38.
    Zhang, S., Liu, D.: Multisoliton solutions of a (2+1)-dimensional variable-coefficient Toda lattice equation via Hirota’s bilinear method. Can. J. Phys. 92, 184–190 (2014)CrossRefGoogle Scholar
  39. 39.
    Zhang, S., Cai, B.: Multi-soliton solutions of a variable-coefficient KdV hierarchy. Nonlin. Dyn. 78, 1593–1600 (2014)CrossRefMathSciNetGoogle Scholar
  40. 40.
    Liu, Y., Gao, Y.T., Sun, Z.Y., Yu, X.: Multi-soliton solutions of the forced variable-coefficient extended Korteweg–de Vries equation arisen in fluid dynamics of internal solitary waves. Nonlin. Dyn. 66, 575–587 (2011)CrossRefMathSciNetMATHGoogle Scholar
  41. 41.
    Zabusky, N.J., Kruskal, M.D.: Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)CrossRefMATHGoogle Scholar
  42. 42.
    Chen, H.H., Liu, C.S.: Solitons in nonuniform media. Phys. Rev. Lett. 37, 693–697 (1976)CrossRefMathSciNetGoogle Scholar
  43. 43.
    Hirota, R., Satsuma, J.: \(N\)-soliton solutions of the K-dV equation with loss and nonuniformity terms. J. Phys. Soc. Jpn. Lett. 41, 2141–2142 (1976)CrossRefGoogle Scholar
  44. 44.
    Calogero, F., Degasperis, A.: Coupled nonlinear evolution equations solvable via the inverse spectral transform, and solitons that come back: the boomeron. Lett. Nuovo Cimento 16, 425–433 (1976)CrossRefMathSciNetGoogle Scholar
  45. 45.
    Chen, H.H., Liu, C.S.: Nonlinear wave and soliton propagation in media with arbitrary inhomogeneities. Phys. Fluids. 21, 377–380 (1978)CrossRefMATHGoogle Scholar
  46. 46.
    Serkin, V.N., Hasegawa, A.: Novel soliton solutions of the nonlinear Schrödinger equation model. Phys. Rev. Lett. 85, 4502–4505 (2000)CrossRefGoogle Scholar
  47. 47.
    Serkin, V.N., Belyaeva, T.L.: The Lax representation in the problem of soliton management. Quant. Electron. 31, 1007–1015 (2001)CrossRefGoogle Scholar
  48. 48.
    Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous solitons in external potentials. Phys. Rev. Lett. 98, 074102 (2007)CrossRefGoogle Scholar
  49. 49.
    Serkin, A., Hasegawa, V.N., Belyaeva, T.L.: Solitary waves in nonautonomous nonlinear and dispersive systems: nonautonomous solitons. J. Mod. Opt. 57, 1456–1472 (2010)CrossRefMATHGoogle Scholar
  50. 50.
    Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous matter-wave solitons near the Feshbach resonance. Phys. Rev. A 81, 023610 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsBohai UniversityJinzhouPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsKashgar UniversityKashgarPeople’s Republic of China

Personalised recommendations