
SIGMA, 2014, Volume 10, 066, 13 pages
(Mi sigma931)




This article is cited in 1 scientific paper (total in 1 paper)
NonPoint Invertible Transformations and Integrability of Partial Difference Equations
Sergey Ya. Startsev^{} ^{} Ufa Institute of Mathematics, Russian Academy of Sciences,
112 Chernyshevsky Str., Ufa, 450077, Russia
Abstract:
This article is devoted to the partial difference quadgraph equations that can be represented in the form $\varphi (u(i+1,j),u(i+1,j+1))=\psi (u(i,j),u(i,j+1))$, where the map $(w,z) \rightarrow (\varphi(w,z),\psi(w,z))$ is injective. The transformation $v(i,j)=\varphi (u(i,j),u(i,j+1))$ relates any of such equations to a quadgraph equation. It is proved that this transformation maps Darboux integrable equations of the above form into Darboux integrable equations again and decreases the orders of the transformed integrals by one in the $j$direction. As an application of this fact, the Darboux integrable equations possessing integrals of the second order in the $j$direction are described under an additional assumption. The transformation also maps symmetries of the original equations into symmetries of the transformed equations (i.e.preserves the integrability in the sense of the symmetry approach) and acts as a difference substitution for symmetries of a special form. The latter fact allows us to derive necessary conditions of Darboux integrability for the equations defined in the first sentence of the abstract.
Keywords:
quadgraph equation; nonpoint transformation; Darboux integrability; higher symmetry; difference substitution; discrete Liouville equation.
DOI:
https://doi.org/10.3842/SIGMA.2014.066
Full text:
PDF file (377 kB)
Full text:
http://www.emis.de/journals/SIGMA/2014/066/
References:
PDF file
HTML file
Bibliographic databases:
ArXiv:
1311.2240
MSC: 39A14; 37K05; 37K10; 37K35 Received: November 10, 2013; in final form June 11, 2014; Published online June 17, 2014
Language:
Citation:
Sergey Ya. Startsev, “NonPoint Invertible Transformations and Integrability of Partial Difference Equations”, SIGMA, 10 (2014), 066, 13 pp.
Citation in format AMSBIB
\Bibitem{Sta14}
\by Sergey~Ya.~Startsev
\paper NonPoint Invertible Transformations and Integrability of Partial Difference Equations
\jour SIGMA
\yr 2014
\vol 10
\papernumber 066
\totalpages 13
\mathnet{http://mi.mathnet.ru/sigma931}
\crossref{https://doi.org/10.3842/SIGMA.2014.066}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3226984}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000338299700001}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2s2.084902577718}
Linking options:
http://mi.mathnet.ru/eng/sigma931 http://mi.mathnet.ru/eng/sigma/v10/p66
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:

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

Number of views: 
This page:  108  Full text:  35  References:  25 
