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

摘要

A new integrable nonautonomous nonlinear ordinary difference equation ispresented which can be considered to be a discrete analogue of the Painleve Vequation. Its derivation is based on the similarity reduction on thetwo-dimensional lattice of integrable partial difference equations of KdV type.The new equation which is referred to as GDP (generalised discrete Painleveequation) contains various ``discrete Painleve equations'' as subcases forspecial values/limits of the parameters, some of which were already givenbefore in the literature. The general solution of the GDP can be expressed interms of Painleve VI (PVI) transcendents. In fact, continuous PVI emerges asthe equation obeyed by the solutions of the discrete equation in terms of thelattice parameters rather than the lattice variables that label the latticesites. We show that the bilinear form of PVI is embedded naturally in thelattice systems leading to the GDP. Further results include the establishmentof Baecklund and Schlesinger transformations for the GDP, the correspondingisomonodromic deformation problem, and the self-duality of its bilinear scheme.