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

This article is cited in 6 scientific papers (total in 6 papers)

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

Bibliographic databases:

Received: 13.01.2004
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
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\transl
\jour Theoret. and Math. Phys.
\yr 2004
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\pages 1542--1561
\crossref{https://doi.org/10.1023/B:TAMP.0000046562.61970.ef}
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    Citing articles on Google Scholar: Russian citations, English citations
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    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.  mathnet  crossref  mathscinet  zmath
    2. Svinin, AK, “Reductions of integrable lattices”, Journal of Physics A-Mathematical and Theoretical, 41:31 (2008), 315205  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    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  crossref  zmath  adsnasa  isi  scopus  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    6. Svinin A.K., “On Some Classes of Discrete Polynomials and Ordinary Difference Equations”, J. Phys. A-Math. Theor., 47:15 (2014), 155201  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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