dorsal/arxiv
View SchemaOn Discrete Painleve Equations Associated with the Lattice KdV Systems and the Painleve VI Equation
| Authors | F. W. Nijhoff, A. Ramani, B. Grammaticos, Y. Ohta |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9812011 |
| URL | https://arxiv.org/abs/solv-int/9812011 |
Abstract
A new integrable nonautonomous nonlinear ordinary difference equation is presented which can be considered to be a discrete analogue of the Painleve V equation. Its derivation is based on the similarity reduction on the two-dimensional lattice of integrable partial difference equations of KdV type. The new equation which is referred to as GDP (generalised discrete Painleve equation) contains various ``discrete Painleve equations'' as subcases for special values/limits of the parameters, some of which were already given before 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 Baecklund and Schlesinger transformations for the GDP, the corresponding isomonodromic deformation problem, and the self-duality of its bilinear scheme.
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"abstract": "A new integrable nonautonomous nonlinear ordinary difference equation is\npresented which can be considered to be a discrete analogue of the Painleve V\nequation. Its derivation is based on the similarity reduction on the\ntwo-dimensional lattice of integrable partial difference equations of KdV type.\nThe new equation which is referred to as GDP (generalised discrete Painleve\nequation) contains various ``discrete Painleve equations\u0027\u0027 as subcases for\nspecial values/limits of the parameters, some of which were already given\nbefore in the literature. The general solution of the GDP can be expressed in\nterms of Painleve VI (PVI) transcendents. In fact, continuous PVI emerges as\nthe equation obeyed by the solutions of the discrete equation in terms of the\nlattice parameters rather than the lattice variables that label the lattice\nsites. We show that the bilinear form of PVI is embedded naturally in the\nlattice systems leading to the GDP. Further results include the establishment\nof Baecklund and Schlesinger transformations for the GDP, the corresponding\nisomonodromic deformation problem, and the self-duality of its bilinear scheme.",
"arxiv_id": "solv-int/9812011",
"authors": [
"F. W. Nijhoff",
"A. Ramani",
"B. Grammaticos",
"Y. Ohta"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "On Discrete Painleve Equations Associated with the Lattice KdV Systems and the Painleve VI Equation",
"url": "https://arxiv.org/abs/solv-int/9812011"
},
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