RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



TMF:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


TMF, 2018, Volume 196, Number 2, Pages 294–312 (Mi tmf9471)  

This article is cited in 3 scientific papers (total in 3 papers)

A direct algorithm for constructing recursion operators and Lax pairs for integrable models

I. T. Habibullinab, A. R. Khakimovaab

a Institution of Russian Academy of Sciences Institute of Mathematics with Computer Center, Ufa, Russia
b Bashkir State University, Ufa, Russia

Abstract: We suggest an algorithm for seeking recursion operators for nonlinear integrable equations. We find that the recursion operator $R$ can be represented as a ratio of the form $R=L_1^{-1}L_2$, where the linear differential operators $L_1$ and $L_2$ are chosen such that the ordinary differential equation $(L_2-\lambda L_1)U=0$ is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter $\lambda\in\mathbb{C}$. To construct the operator $L_1$, we use the concept of an invariant manifold, which is a generalization of a symmetry. To seek $L_2$, we then take an auxiliary linear equation related to the linearized equation by a Darboux transformation. It is remarkable that the equation $L_1\widetilde U=L_2U$ defines a Bäcklund transformation mapping a solution $U$ of the linearized equation to another solution $\widetilde U$ of the same equation. We discuss the connection of the invariant manifold with the Lax pairs and the Dubrovin equations.

Keywords: Lax pair, integrable chain, higher symmetry, invariant manifold, recursion operator.

DOI: https://doi.org/10.4213/tmf9471

Full text: PDF file (507 kB)
First page: PDF file
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 2018, 196:2, 1200–1216

Bibliographic databases:

Received: 29.09.2017

Citation: I. T. Habibullin, A. R. Khakimova, “A direct algorithm for constructing recursion operators and Lax pairs for integrable models”, TMF, 196:2 (2018), 294–312; Theoret. and Math. Phys., 196:2 (2018), 1200–1216

Citation in format AMSBIB
\Bibitem{HabKha18}
\by I.~T.~Habibullin, A.~R.~Khakimova
\paper A~direct algorithm for constructing recursion operators and Lax pairs for integrable models
\jour TMF
\yr 2018
\vol 196
\issue 2
\pages 294--312
\mathnet{http://mi.mathnet.ru/tmf9471}
\crossref{https://doi.org/10.4213/tmf9471}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2018TMP...196.1200H}
\elib{http://elibrary.ru/item.asp?id=35276545}
\transl
\jour Theoret. and Math. Phys.
\yr 2018
\vol 196
\issue 2
\pages 1200--1216
\crossref{https://doi.org/10.1134/S004057791808007X}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000443722200007}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85052687266}


Linking options:
  • http://mi.mathnet.ru/eng/tmf9471
  • https://doi.org/10.4213/tmf9471
  • http://mi.mathnet.ru/eng/tmf/v196/i2/p294

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Habibullin I.T. Khakimova A.R., “On the Recursion Operators For Integrable Equations”, J. Phys. A-Math. Theor., 51:42 (2018), 425202  crossref  isi  scopus
    2. A. R. Khakimova, “On description of generalized invariant manifolds for nonlinear equations”, Ufa Math. J., 10:3 (2018), 106–116  mathnet  crossref  isi
    3. Zhang Zh.-Y., “An Upper Order Bound of the Invariant Manifold in Lax Pairs of a Nonlinear Evolution Partial Differential Equation”, J. Phys. A-Math. Theor., 52:26 (2019), 265202  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
    Number of views:
    This page:119
    References:10
    First page:11

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019