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 TMF, 2004, Volume 141, Number 2, Pages 243–266 (Mi tmf124)

Invariant Submanifolds of the Darboux–Kadomtsev–Petviashvili Chain and an Extension of the Discrete Kadomtsev–Petviashvili Hierarchy

A. K. Svinin

Institute of System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences

Abstract: We investigate invariant submanifolds of the so-called Darboux–Kadomtsev–Petviashvili chain. We show that restricting the dynamics to a class of invariant submanifolds yields an extension of the discrete Kadomtsev–Petviashvili hierarchy and intersections of invariant submanifolds yield the Lax description of a wide class of differential-difference systems. We consider self-similar reductions. We show that self-similar substitutions result in purely discrete equations that depend on a finite set of parameters and in equations determining deformations w.r.t. these parameters. We present examples. In particular, we show that the well-known first discrete Painlevé equation corresponds to the Volterra chain hierarchy. We derive the equations naturally generalizing the first discrete Painlevé equation in the sense that all of them become the first Painlevé equation in the continuum limit.

Keywords: discrete Kadomtsev–Petviashvili hierarchy, invariant submanifolds, Darboux map

DOI: https://doi.org/10.4213/tmf124

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English version:
Theoretical and Mathematical Physics, 2004, 141:2, 1542–1561

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Revised: 22.03.2004

Citation: A. K. Svinin, “Invariant Submanifolds of the Darboux–Kadomtsev–Petviashvili Chain and an Extension of the Discrete Kadomtsev–Petviashvili Hierarchy”, TMF, 141:2 (2004), 243–266; Theoret. and Math. Phys., 141:2 (2004), 1542–1561

Citation in format AMSBIB
\Bibitem{Svi04} \by A.~K.~Svinin \paper Invariant Submanifolds of the Darboux--Kadomtsev--Petviashvili Chain and an~Extension of the Discrete Kadomtsev--Petviashvili Hierarchy \jour TMF \yr 2004 \vol 141 \issue 2 \pages 243--266 \mathnet{http://mi.mathnet.ru/tmf124} \crossref{https://doi.org/10.4213/tmf124} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2120226} \zmath{https://zbmath.org/?q=an:1178.37093} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2004TMP...141.1542S} \transl \jour Theoret. and Math. Phys. \yr 2004 \vol 141 \issue 2 \pages 1542--1561 \crossref{https://doi.org/10.1023/B:TAMP.0000046562.61970.ef} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000225778500006} 

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• https://doi.org/10.4213/tmf124
• http://mi.mathnet.ru/eng/tmf/v141/i2/p243

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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. Andrei K. Svinin, “Integrable Discrete Equations Derived by Similarity Reduction of the Extended Discrete KP Hierarchy”, SIGMA, 2 (2006), 005, 11 pp.
2. Svinin, AK, “Reductions of integrable lattices”, Journal of Physics A-Mathematical and Theoretical, 41:31 (2008), 315205
3. Svinin, AK, “On some class of reductions for the Itoh-Narita-Bogoyavlenskii lattice”, Journal of Physics A-Mathematical and Theoretical, 42:45 (2009), 454021
4. Svinin A.K., “On some integrable lattice related by the Miura-type transformation to the Itoh-Narita-Bogoyavlenskii lattice”, J. Phys. A: Math. Theor., 44:46 (2011), 465210
5. Svinin A.K., “On some class of homogeneous polynomials and explicit form of integrable hierarchies of differential-difference equations”, J. Phys. A: Math. Theor., 44:16 (2011), 165206
6. Svinin A.K., “On Some Classes of Discrete Polynomials and Ordinary Difference Equations”, J. Phys. A-Math. Theor., 47:15 (2014), 155201
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