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Discrete Painleve equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three different types of equations according to whether the coefficient functions are linear, exponential or elliptic functions of $n$. In this paper, we focus on the elliptic type and give a review of the construction of such equations on the $E_8$ lattice. The first such construction was given by Sakai cite{SakaiH2001:MR1882403}. We focus on recent developments giving rise to more examples of elliptic discrete Painleve equations.
In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painleve equations, with a particular emphasis on the discrete Painleve equations. The theory is controlled by the geometry of certain
We show some classes of higher order partial difference equations admitting a zero-curvature representation and generalizing lattice potential KdV equation. We construct integrable hierarchies which, as we suppose, yield generalized symmetries for ob
In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre Unitary ensemble, from the point of view of Sakais geometric theory of Painleve equations. On one hand, this gives us one more detail
We study higher order KdV equations from the GL(2,$mathbb{R}$) $cong$ SO(2,1) Lie group point of view. We find elliptic solutions of higher order KdV equations up to the ninth order. We argue that the main structure of the trigonometric/hyperbolic/el
A theoretical foundation for a generalization of the elliptic difference Painleve equation to higher dimensions is provided in the framework of birational Weyl group action on the space of point configurations in general position in a projective spac