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TMF, 2008, Volume 156, Number 2, Pages 207–219 (Mi tmf6241)  

This article is cited in 2 scientific papers (total in 2 papers)

Darboux-integrable discrete systems

V. L. Vereshchagin

Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences

Abstract: We extend Laplace's cascade method to systems of discrete “hyperbolic” equations of the form $u_{i+1,j+1}=f(u_{i+1,j},u_{i,j+1},u_{i,j})$, where $u_{ij}$ is a member of a sequence of unknown vectors, $i,j\in\mathbb Z$. We introduce the notion of a generalized Laplace invariant and the associated property of the system being “Liouville.” We prove several statements on the well-definedness of the generalized invariant and on its use in the search for solutions and integrals of the system. We give examples of discrete Liouville-type systems.

Keywords: Laplace's cascade method, Darboux integrability, nonlinear chain

DOI: https://doi.org/10.4213/tmf6241

Full text: PDF file (459 kB)
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English version:
Theoretical and Mathematical Physics, 2008, 156:2, 1142–1153

Bibliographic databases:

Received: 16.05.2007
Revised: 09.07.2007

Citation: V. L. Vereshchagin, “Darboux-integrable discrete systems”, TMF, 156:2 (2008), 207–219; Theoret. and Math. Phys., 156:2 (2008), 1142–1153

Citation in format AMSBIB
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  • https://doi.org/10.4213/tmf6241
  • http://mi.mathnet.ru/eng/tmf/v156/i2/p207

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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. V. L. Vereshchagin, “Discrete Toda lattices and the Laplace method”, Theoret. and Math. Phys., 160:3 (2009), 1229–1237  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. Gubbiotti G. Levi D. Scimiterna Ch., “On Partial Differential and Difference Equations With Symmetries Depending on Arbitrary Functions”, Acta Polytech., 56:3 (2016), 193–201  crossref  isi  scopus  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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