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

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

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

Bibliographic databases:

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} 

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

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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
2. A. R. Khakimova, “On description of generalized invariant manifolds for nonlinear equations”, Ufa Math. J., 10:3 (2018), 106–116
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
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