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Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Studies in Applied Mathematics
( Vol. 106 NO.3 / (2001))
On Discrete Painleve Equations Associated with the Lattice KdV Systems and the Painleve VI Equation
F. W. Nijhoff, A. Ramani, . Grammaticos, Y. Ohta,
Pages. 261-314
Abstract A new integrable nonautonomous nonlinear ordinary difference
equation is presented that can be considered to be a discrete
analogue of the Painleve V equation. It 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
Painleve equation (GDP), contains various "discrete Painleve
equations" as subcases for special values/limits of the
parameters, some of which have already bee given in the
literature. The general solution of the GDP can be expressed in
terms of Painleve 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
Backlund and Schlesinger transformations for the GDP, the
corresponding isomonodromic deformation problem, and the
self-duality of its bilinear scheme.
Contents 1. Introduction 2. Derivation: Infinite-matrix scheme 3. The
lattice Painleve system 4. Integrability aspects 5. Schlesinger
transformations and self-duality 6. Conclusions
Key words
Mathmatical Subject Classification