On Discrete Painlevé Equations Associated with the Lattice KdV Systems and the Painlevé VI Equation

A new integrable nonautonomous nonlinear ordinary difference equation is presented that can be considered to be a discrete analogue of the Painlevé V equation. Its derivation is based on the similarity reduction on the two-dimensional lattice of integrable partial differential equations of Korteweg–de Vries (KdV) type. The new equation, which is referred to as generalized discrete Painlevé equation (GDP), contains various "discrete Painlevé equations" as subcases for special values/limits of the parameters, some of which have already been given in the literature. The general solution of the GDP can be expressed in terms of Painlevé VI (PVI) transcendents. In fact, continuous PVI emerges as the equation obeyed by the solutions of the discrete equation in terms of the lattice parameters rather than the lattice variables that label the lattice sites. We show that the bilinear form of PVI is embedded naturally in the lattice systems leading to the GDP. Further results include the establishment of Bäcklund and Schlesinger transformations for the GDP, the corresponding isomonodromic deformation problem, and the self-duality of its bilinear scheme.