IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v191y2022icp157-167.html
   My bibliography  Save this article

Exact solitary wave and periodic-peakon solutions of the complex Ginzburg–Landau equation: Dynamical system approach

Author

Listed:
  • Xu, Guoan
  • Zhang, Yi
  • Li, Jibin

Abstract

Using the bifurcation theory of the planar dynamical system, we study the exact solutions of the complex Ginzburg–Landau equation which is a popular model in mathematical physics. All possible exact explicit parametric representations of traveling wave solutions are given under different parameter conditions, including the solitary wave solutions, periodic wave solutions, compacton solutions pseudo-peakon solutions and periodic peakon solutions. In more general parametric conditions, all possible solutions are caught in one dragnet.

Suggested Citation

  • Xu, Guoan & Zhang, Yi & Li, Jibin, 2022. "Exact solitary wave and periodic-peakon solutions of the complex Ginzburg–Landau equation: Dynamical system approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 191(C), pages 157-167.
    • Handle: RePEc:eee:matcom:v:191:y:2022:i:c:p:157-167
    DOI: 10.1016/j.matcom.2021.08.007
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475421002846
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2021.08.007?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Kudryashov, Nikolay A., 2020. "Highly dispersive solitary wave solutions of perturbed nonlinear Schrödinger equations," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    2. Mancas, Stefan C. & Choudhury, S.Roy, 2007. "The complex cubic–quintic Ginzburg–Landau equation: Hopf bifurcations yielding traveling waves," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 74(4), pages 281-291.
    3. Kudryashov, Nikolay A., 2020. "First integrals and general solution of the complex Ginzburg-Landau equation," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    4. Djob, Roger Bertin & Kenfact-Jiotsa, Aurelien & Govindarajan, A., 2020. "Non-Lagrangian approach for coupled complex Ginzburg-Landau systems with higher order-dispersion," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    5. Kudryashov, Nikolai A., 2005. "Simplest equation method to look for exact solutions of nonlinear differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 24(5), pages 1217-1231.
    6. Shi, Dongyang & Liu, Qian, 2020. "Superconvergence analysis of a two grid finite element method for Ginzburg–Landau equation," Applied Mathematics and Computation, Elsevier, vol. 365(C).
    7. Qiu, Yunli & Malomed, Boris A. & Mihalache, Dumitru & Zhu, Xing & Zhang, Li & He, Yingji, 2020. "Soliton dynamics in a fractional complex Ginzburg-Landau model," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    8. Zhu, Wenjing & Xia, Yonghui & Bai, Yuzhen, 2020. "Traveling wave solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity," Applied Mathematics and Computation, Elsevier, vol. 382(C).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Chaudry Masood Khalique & Karabo Plaatjie, 2021. "Symmetry Methods and Conservation Laws for the Nonlinear Generalized 2D Equal-Width Partial Differential Equation of Engineering," Mathematics, MDPI, vol. 10(1), pages 1-17, December.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kudryashov, Nikolay A., 2020. "First integrals and general solution of the complex Ginzburg-Landau equation," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    2. Ekici, Mehmet, 2022. "Kinky breathers, W-shaped and multi-peak soliton interactions for Kudryashov's quintuple power-law coupled with dual form of non-local refractive index structure," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    3. Kudryashov, Nikolay A., 2020. "Optical solitons of model with integrable equation for wave packet envelope," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    4. Kumar, Vikas & Biswas, Anjan & Ekici, Mehmet & Moraru, Luminita & Alzahrani, Abdullah Khamis & Belic, Milivoj R., 2021. "Time–dependent coupled complex short pulse equation: Invariant analysis and complexitons," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    5. Merabti, Abdelouahab & Triki, Houria & Azzouzi, Faiçal & Zhou, Qin & Biswas, Anjan & Liu, Wenjun & Alzahrani, Abdullah Kamis & EL-Akrmi, Abdessetar, 2020. "Propagation properties of chirped optical similaritons with dual-power law nonlinearity," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    6. Innocent Simbanefayi & Chaudry Masood Khalique, 2020. "Group Invariant Solutions and Conserved Quantities of a (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation," Mathematics, MDPI, vol. 8(6), pages 1-20, June.
    7. Fahmy, E.S., 2008. "Travelling wave solutions for some time-delayed equations through factorizations," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 1209-1216.
    8. Li, Pengfei & Malomed, Boris A. & Mihalache, Dumitru, 2020. "Symmetry breaking of spatial Kerr solitons in fractional dimension," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    9. Mustafa Inc & Rubayyi T. Alqahtani & Ravi P. Agarwal, 2023. "W-Shaped Bright Soliton of the (2 + 1)-Dimension Nonlinear Electrical Transmission Line," Mathematics, MDPI, vol. 11(7), pages 1-13, April.
    10. Chaudry Masood Khalique & Karabo Plaatjie, 2021. "Symmetry Methods and Conservation Laws for the Nonlinear Generalized 2D Equal-Width Partial Differential Equation of Engineering," Mathematics, MDPI, vol. 10(1), pages 1-17, December.
    11. Yang, Lijuan & Du, Xianyun & Yang, Qiongfen, 2016. "New variable separation solutions to the (2 + 1)-dimensional Burgers equation," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 1271-1275.
    12. Zayed, E.M.E. & Alurrfi, K.A.E., 2016. "Extended auxiliary equation method and its applications for finding the exact solutions for a class of nonlinear Schrödinger-type equations," Applied Mathematics and Computation, Elsevier, vol. 289(C), pages 111-131.
    13. Kengne, Emmanuel, 2021. "Modulational instability and soliton propagation in an alternate right-handed and left-handed multi-coupled nonlinear dissipative transmission network," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    14. Andrei D. Polyanin, 2019. "Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions," Mathematics, MDPI, vol. 7(5), pages 1-19, April.
    15. Kudryashov, Nikolay A. & Ryabov, Pavel N., 2014. "Exact solutions of one pattern formation model," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 1090-1093.
    16. Zdravković, S. & Zeković, S. & Bugay, A.N. & Petrović, J., 2021. "Two component model of microtubules and continuum approximation," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    17. Navickas, Z. & Ragulskis, M. & Telksnys, T., 2016. "Existence of solitary solutions in a class of nonlinear differential equations with polynomial nonlinearity," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 333-338.
    18. Zayed, Elsayed M.E. & Alngar, Mohamed E.M. & Biswas, Anjan & Asma, Mir & Ekici, Mehmet & Alzahrani, Abdullah K. & Belic, Milivoj R., 2020. "Optical solitons and conservation laws with generalized Kudryashov’s law of refractive index," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    19. Zeng, Liangwei & Mihalache, Dumitru & Malomed, Boris A. & Lu, Xiaowei & Cai, Yi & Zhu, Qifan & Li, Jingzhen, 2021. "Families of fundamental and multipole solitons in a cubic-quintic nonlinear lattice in fractional dimension," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    20. He, Shangling & Malomed, Boris A. & Mihalache, Dumitru & Peng, Xi & Yu, Xing & He, Yingji & Deng, Dongmei, 2021. "Propagation dynamics of abruptly autofocusing circular Airy Gaussian vortex beams in the fractional Schrödinger equation," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:matcom:v:191:y:2022:i:c:p:157-167. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.