Jump to ContentJump to Main Navigation
Show Summary Details
More options …

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Chen, Xi / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

IMPACT FACTOR 2017: 1.162

CiteScore 2017: 1.41

SCImago Journal Rank (SJR) 2017: 0.382
Source Normalized Impact per Paper (SNIP) 2017: 0.636

Mathematical Citation Quotient (MCQ) 2017: 0.12

See all formats and pricing
More options …
Volume 19, Issue 3-4


Lax Integrability and Exact Solutions of a Variable-Coefficient and Nonisospectral AKNS Hierarchy

Sheng Zhang / Siyu Hong
Published Online: 2018-04-19 | DOI: https://doi.org/10.1515/ijnsns-2016-0191


In this paper, a variable-coefficient and nonisospectral Ablowitz–Kaup–Newell–Segur (vcniAKNS) hierarchy with Lax integrability is constructed by embedding a finite number of differentiable and time-dependent functions into the well-known AKNS spectral problem and its time evolution equation. In the framework of inverse scattering transform method with time-varying spectral parameter, the constructed vcniAKNS hierarchy is solved exactly. As a result, exact solutions and their reduced n-soliton solutions of the vcniAKNS hierarchy are obtained. It is graphically shown that the parity of an embedded time-dependent function has connection with the symmetrical characteristics of the spatial structures and singular points of the obtained one-soliton solutions.

Keywords: lax integrability; exact solution; soliton solution; vcniAKNS hierarchy; inverse scattering transform

PACS: 05.45.Yv; 04.20.Jb; 02.30.Jr


  • [1]

    Garder C. S., Greene J. M., Kruskal M. D. and Miura R. M., Method for solving the Korteweg–de Vries equation, Phys. Rev. Lett. 19 (1965), 1095–1097.Google Scholar

  • [2]

    Hirota R., Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971), 1192–1194.CrossrefGoogle Scholar

  • [3]

    Weiss J., Tabor M. and Carnevale G., The Painlevé property for partial differential equations, J. Math. Phys. 24 (1983), 522–526.

  • [4]

    Wang M. L., Exact solutions for a compound KdV–Burgers equation, Phys. Lett. A 213 (1996), 279–287.Google Scholar

  • [5]

    Fan E. G., Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems, Phys. Lett. A 300 (2002), 243–249.

  • [6]

    He J. H. and Wu X. H., Exp-function method for nonlinear wave equations, Chaos Soliton. Fract. 30 (2006), 700–708.Google Scholar

  • [7]

    Zhang S. and 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 (2007), 227–248.

  • [8]

    Yomba E., The modified extended Fan sub-equation method and its application to the (2+1)-dimensional Broer–Kaup–Kupershmidt equation, Chaos Soliton. Fract. 27 (2006), 187–196.

  • [9]

    Zhang S. and Liu D. D., The third kind of Darboux transformation and multisoliton solutions for generalized Broer–Kaup equations, Turk. J. Phys. 39 (2015), 165–177.Web of ScienceCrossrefGoogle Scholar

  • [10]

    Dai C. Q. and Wang Y. Y., Controllable combined Peregrine soliton and Kuznetsov–Ma soliton in PT-symmetric nonlinear couplers with gain and loss, Dyn Nonlinear. 80 (2015), 715–721.

  • [11]

    Dai C. Q., Fan Y., Zhou G. Q., Zheng J. and Cheng L., Vector spatiotemporal localized structures in (3+1)-dimensional strongly nonlocal nonlinear media, Nonlinear Dyn. 86 (2016), 999–1005.

  • [12]

    Zhang S. and Xia T. C., Variable-coefficient Jacobi elliptic function expansion method for (2+1)-dimensional Nizhnik–Novikov–Vesselov equations, Appl. Math. Comput. 218 (2011), 1308–1316.

  • [13]

    Zhang N. and Xia T. C., A hierarchy of lattice soliton equations associated with a new discrete eigenvalue problem and Darboux transformations, Int. J. Nonlinear Sci. Numer. Simul. 16 (2015), 301–306.

  • [14]

    Dai C. Q., Wang X. G. and Zhou G. Q., Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials, Phys. Rev. A 89 (2014), 013834.Web of ScienceGoogle Scholar

  • [15]

    Dai C. Q., Chen R. P., Wang Y. Y. and Fan Y., Dynamics of light bullets in inhomogeneous cubic-quintic-septimal nonlinear media with PT-symmetric potentials, Nonlinear Dyn. 87 (2017), 1675–1683.Web of ScienceCrossrefGoogle Scholar

  • [16]

    Kong L. Q. and Dai C. Q., Some discussions about variable separation of nonlinear models using Riccati equation expansion method, Nonlinear Dyn. 81 (2015), 1553–1561.CrossrefWeb of ScienceGoogle Scholar

  • [17]

    Nachman A. I. and Ablowitz M. J., A multidimensional inverse scattering method, Stud. Appl. Math. 71 (1984), 243–250.CrossrefGoogle Scholar

  • [18]

    Chan W. L. and Li K. S., Nonpropagating solitons of the variable coefficient and nonisospectral Korteweg-de Vries equation, J. Math. Phys. 30 (1989), 2521–2526.

  • [19]

    Xu B. Z. and Zhao S. Q., Inverse scattering transformation for the variable coefficient sine-Gordon type equation, Appl. Math. JCU 9B (1994), 331–337.Google Scholar

  • [20]

    Zeng Y. B., Ma W. X. and Lin R. L., Integration of the soliton hierarchy with selfconsistent sources, J. Math. Phys. 41 (2000), 5453–5489.

  • [21]

    Biondini G. and Kovacic G., Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions, J. Math. Phys. 55 (2014), 031506.

  • [22]

    Chakravarty S., Prinari B. and Ablowitz M. J., Inverse scattering transform for 3-level coupled Maxwell-Bloch equations with inhomogeneous broadening, Physica D 278–279 (2014), 58–78.

  • [23]

    Zhang S. and Gao X. D., Mixed spectral AKNS hierarchy from linear isospectral problem and its exact solutions, Open Phys. 13 (2015), 310–322.Web of ScienceGoogle Scholar

  • [24]

    Zhang S., Xu B. and Zhang H. Q., Exact solutions of a KdV equation hierarchy with variable coefficients, Int. J. Comput. Math. 91 (2014), 1601–1616.Web of ScienceCrossrefGoogle Scholar

  • [25]

    Zhang S. and Wang D., Variable-coefficient nonisospectral Toda lattice hierarchy and its exact solutions, Pramana–J. Phys. 85 (2015), 1143–1156.Web of ScienceCrossrefGoogle Scholar

  • [26]

    Zhang S. and Gao X. D., Exact solutions and dynamics of a generalized AKNS equations associated with the nonisospectral depending on exponential function, J. Nonlinear Sci. Appl. 19 (2016), 4529–4541.

  • [27]

    Gao X. D. and Zhang S., Time-dependent-coefficient AKNS hierarchy and its exact multi-soliton solutions, Int. J. Appl. Sci. Math. 3 (2016), 72–75.

  • [28]

    Chen H. H. and Liu C. S., Solitons in nonuniform media, Phys. Rev. Lett. 37 (1976), 693–697.CrossrefGoogle Scholar

  • [29]

    Hirota R. and Satsuma J., N-soliton solutions of the K-dV equation with loss and nonuniformity terms, J. Phys. Soc. Jpn. 41 (1976), 2141–2142.

  • [30]

    Calogero F. and Degasperis A., Coupled nonlinear evolution equations solvable via the inverse spectral transform, and solitons that come back: the boomeron, Lett. al Nuovo Cimento. 16 (1976), 425–433.CrossrefGoogle Scholar

  • [31]

    Chen D. Y., Introduction of Soliton, Science Press, Beijing, 2006.Google Scholar

  • [32]

    Ablowitz M. J. and Clarkson P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.Google Scholar

  • [33]

    Calogreo F. and Degasperis A., Exact solutions via the spetr transform method for solving nonlinear evolustions, Lett. al Nuovo Cimento. 22 (1978), 131–137.

  • [34]

    Calogreo F. and Degasperis A., Extension of the spectral transform method for solving nonlinear evolustions, Lett. al Nuovo Cimento 22 (1978), 263–269.

  • [35]

    Calogreo F. and Degasperis A., Exact solution via the spectral transform of a generalization with linearly x-dependent coefficients of the modified Korteweg–de Vries equation, Lett. al Nuovo Cimento 22 (1978), 270–273.

  • [36]

    Li Y., A class of evolution equations and the spectral deformation, Sci. China Ser. A: Math. 25 (1982), 911–917.

  • [37]

    Serkin V. N. and Hasegawa A., Novel soliton solutions of the nonlinear Schrödinger equation model, Phys. Review Lett. 85 (2000), 4502–4505.

  • [38]

    Serkin V. N. and Belyaeva T. L., The Lax representation in the problem of soliton management, Electron Quantum. 31 (2001), 1007–1015.

  • [39]

    Serkin V. N., A. Hasegawa and Belyaeva T. L., Nonautonomous solitons in external potentials, Phys. Rev. Lett. 98 (2007), 074102.Google Scholar

  • [40]

    Serkin V. N., Hasegawa A. and Belyaeva T. L., Solitary waves in nonautonomous nonlinear and dispersive systems: nonautonomous solitons, Mod J.. Optic. 57 (2010), 1456–1472.Web of ScienceGoogle Scholar

  • [41]

    Serkin V. N., A. Hasegawa and Belyaeva T. L., Nonautonomous matter-wave solitons near the Feshbach resonance, Phys. Rev. A 81 (2010), 023610.Web of ScienceGoogle Scholar

  • [42]

    Zhang S. and Liu D., Multisoliton solutions of a (2+1)-dimensional variable-coefficient Toda lattice equation via Hirota’s bilinear method, Can. J. Phys. 92 (2014), 184–190.

  • [43]

    Zhang S. and Cai B., Multi-soliton solutions of a variable-coefficient KdV hierarchy, Nonlinear Dyn. 78 (2014), 1593–1600.Web of ScienceCrossrefGoogle Scholar

  • [44]

    Liu Y., Gao Y. T., Sun Z. Y. and Yu X., Multi-soliton solutions of the forced variable-coefficient extended Korteweg–de Vries equation arisen in fluid dynamics of internal solitary vaves, Dyn Nonlinear. 66 (2011), 575–587.

  • [45]

    Zhang S. and Gao X. D., Exact N-soliton solutions and dynamics of a new AKNS equations with time-dependent coefficients, Nonlinear Dyn. 83 (2016), 1043–1052.

  • [46]

    Zhang S., Tian C. and Qian W. Y., Bilinearization and new multi-soliton solutions for the (4+1)-dimensional Fokas equation, Pramana–J. Phys. 86 (2016), 1259–1267.

About the article

Received: 2016-12-26

Accepted: 2017-11-20

Published Online: 2018-04-19

Published in Print: 2018-06-26

This work was supported by the Natural Science Foundation of China (11547005), the Natural Science Foundation of Liaoning Province of China (20170540007), the Natural Science Foundation of Education Department of Liaoning Province of China (LZ2017002) and Innovative Talents Support Program in Colleges and Universities of Liaoning Province (LR2016021).

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 3-4, Pages 251–262, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2016-0191.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in