IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v151y2021ics096007792100610x.html
   My bibliography  Save this article

Generalized Hermite polynomials for the Burgers hierarchy and point vortices

Author

Listed:
  • Kudryashov, Nikolay A.

Abstract

Rational solutions of equations for the Burgers hierarchy are considered. Using self-similar variables this hierarchy is reduced to the family of nonlinear ordinary differential equations. Then the family is transformed to the hierarchy of non-autonomous linear differential equations by means of the Cole-Hopf formula. This hierarchy is a generalization of the second-order equation for Hermite polynomials. It is shown that every member of the hierarchy for ordinary differential equation has the solution in the form of polynomials. Properties of solutions of generalized Hermite equations in the form the special polynomials are studied. A recursion relation that can be used for finding corresponding polynomials for every member is given. It is proved that the well-known property for Hermite polynomials connecting two polynomials can be used for all polynomials of the generalized Hermite hierarchy. It is shown that the Cole-Hopf transformation is a direct consequence of the differential connection between two special polynomials in the hierarchy of Hermite equations. A derivation of the generalized Tkachenko equations is given for polynomials of the generalized Hermite hierarchy whose roots correspond to point vortices in the background flow.

Suggested Citation

  • Kudryashov, Nikolay A., 2021. "Generalized Hermite polynomials for the Burgers hierarchy and point vortices," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    • Handle: RePEc:eee:chsofr:v:151:y:2021:i:c:s096007792100610x
    DOI: 10.1016/j.chaos.2021.111256
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S096007792100610X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2021.111256?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, Nikolai A., 2009. "Special polynomials associated with rational solutions of some hierarchies," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1447-1462.
    2. Gao, Xin-Yi & Guo, Yong-Jiang & Shan, Wen-Rui, 2021. "Beholding the shallow water waves near an ocean beach or in a lake via a Boussinesq-Burgers system," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
    3. Gupta, A.K. & Ray, S. Saha, 2018. "On the solution of time-fractional KdV–Burgers equation using Petrov–Galerkin method for propagation of long wave in shallow water," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 376-380.
    Full references (including those not matched with items on IDEAS)

    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. Yilong Li & Qiang Xu & Yujie Li & Yuanbei Li & Cong Liu, 2022. "Application of Microbial-Induced Calcium Carbonate Precipitation in Wave Erosion Protection of the Sandy Slope: An Experimental Study," Sustainability, MDPI, vol. 14(20), pages 1-16, October.

    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:chsofr:v:151:y:2021:i:c:s096007792100610x. 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: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    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.