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Ufa Mathematical Journal, 2019, Volume 11, Issue 3, Pages 99–108
DOI: https://doi.org/10.13108/2019-11-3-99
(Mi ufa483)
 

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

On series of Darboux integrable discrete equations on square lattice

R. N. Garifullin, R. I. Yamilov

Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Chenryshevsky str. 112, 450008, Ufa, Russia
References:
Abstract: We present a series of Darboux integrable discrete equations on a square lattice. The equations in the series are numbered by natural numbers $M$. All the equations possess a first order first integral in one of directions of the two-dimensional lattice. The minimal order of a first integral in the other direction is equal to $3M$ for an equation with the number $M$. First integrals in the second direction are defined by a simple general formula depending on the number $M$.
In the cases $M=1,2,3$ we show that the equations are integrable by quadrature. More precisely, we construct their general solutions in terms of the discrete integrals.
We also construct a modified series of Darboux integrable discrete equations having the first integrals of the minimal orders $2$ and $3M-1$ in different directions, where $M$ is the equation number in series. Both first integrals are not obvious in this case.
A few similar series of integrable equations were known before; however, they were of Burgers or sine-Gordon type. A similar series of the continuous hyperbolic type equations was discussed by A.V. Zhiber and V.V. Sokolov in 2001.
Keywords: discrete quad-equation, Darboux integrability, general solution.
Received: 11.06.2019
Bibliographic databases:
Document Type: Article
UDC: 517.547
MSC: 39A14, 39A10, 35L10
Language: English
Original paper language: English
Citation: R. N. Garifullin, R. I. Yamilov, “On series of Darboux integrable discrete equations on square lattice”, Ufa Math. J., 11:3 (2019), 99–108
Citation in format AMSBIB
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\by R.~N.~Garifullin, R.~I.~Yamilov
\paper On series of Darboux integrable discrete equations on square lattice
\jour Ufa Math. J.
\yr 2019
\vol 11
\issue 3
\pages 99--108
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  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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