|
This article is cited in 2 scientific papers (total in 2 papers)
Yamilov's theorem for differential and difference equations
Decio Levia, Miguel A. Rodríguezb a Mathematical and Physical Department,
Roma Tre University,
Via della Vasca Navale, 84,
I00146 Roma, Italy
b Dept. Física Teórica,
Universidad Complutense de Madrid,
Pza. de las Ciencias, 1,
28040 Madrid, Spain
Abstract:
S-integrable scalar evolutionary differential difference equations in 1+1
dimensions have a very particular form described by Yamilov's theorem. We
look for similar results in the case of S-integrable 2-dimensional partial difference equations and 2-dimensional partial differential equations. To do so, on one side we discuss the semi-continuous limit of S-integrable quad equations and on the other, we semi-discretize partial differential equations. For partial differential equations, we show that any equation can be semi-discretized in such a way to satisfy Yamilov's theorem. In the case of partial difference equations, we are not able to find a form of the equation such that its semi-continuous limit always satisfies Yamilov's theorem. So we just present a few examples, in which to get evolutionary equations, we need to carry out a skew limit. We also consider an S-integrable quad equation with non-constant coefficients which in the skew limit satisfies an extended Yamilov's theorem as it has non-constant coefficients. This equation turns out to be a subcase of the Yamilov discretization of the Krichever-Novikov equation with non-constant coefficient, an equation suggested to be integrable by Levi and Yamilov in 1997 and whose integrability has been proved only recently by algebraic entropy. If we do a strait limit, we get non-local evolutionary equations, which show that an extension of Yamilov's theorem may exist in this case.
Keywords:
differential difference equations, continuous and
discrete integrable systems, Yamilov's theorem.
Received: 11.03.2021
Citation:
Decio Levi, Miguel A. Rodríguez, “Yamilov's theorem for differential and difference equations”, Ufimsk. Mat. Zh., 13:2 (2021), 158–165; Ufa Math. J., 13:2 (2021), 152–159
Linking options:
https://www.mathnet.ru/eng/ufa563https://doi.org/10.13108/2021-13-2-152 https://www.mathnet.ru/eng/ufa/v13/i2/p158
|
Statistics & downloads: |
Abstract page: | 97 | Russian version PDF: | 61 | English version PDF: | 14 | References: | 16 |
|