Yamilov's theorem for differential and difference equations

Authors

  • Decio Levi
    Mathematical and Physical Department, Roma Tre University, Via della Vasca Navale, 84, I00146 Roma, Italy
  • MiguelA. Rodríguez
    Dept. Física Teórica, Universidad Complutense de Madrid, Pza. de las Ciencias, 1, 28040 Madrid, Spain

DOI:

https://doi.org/10.13108/2021-13-2-152

Keywords:

differential difference equations, continuous and

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.

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Published

20.06.2021