dorsal/arxiv
View SchemaThe Discrete Painlev\'e I Hierarchy
| Authors | Clio Cresswell, Nalini Joshi |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9710021 |
| URL | https://arxiv.org/abs/solv-int/9710021 |
Abstract
The discrete Painlev\'e I equation (dP$\rm_I$) is an integrable difference equation which has the classical first Painlev\'e equation (P$\rm_I$) as a continuum limit. dP$\rm_I$ is believed to be integrable because it is the discrete isomonodromy condition for an associated (single-valued) linear problem. In this paper, we derive higher-order difference equations as isomonodromy conditions that are associated to the same linear deformation problem. These form a hierarchy that may be compared to hierarchies of integrable ordinary differential equations (ODEs). We strengthen this comparison by continuum limit calculations that lead to equations in the P$\rm_I$ hierarchy. We propose that our difference equations are discrete versions of higher-order Painlev\'e equations.
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"abstract": "The discrete Painlev\\\u0027e I equation (dP$\\rm_I$) is an integrable difference\nequation which has the classical first Painlev\\\u0027e equation (P$\\rm_I$) as a\ncontinuum limit. dP$\\rm_I$ is believed to be integrable because it is the\ndiscrete isomonodromy condition for an associated (single-valued) linear\nproblem. In this paper, we derive higher-order difference equations as\nisomonodromy conditions that are associated to the same linear deformation\nproblem. These form a hierarchy that may be compared to hierarchies of\nintegrable ordinary differential equations (ODEs). We strengthen this\ncomparison by continuum limit calculations that lead to equations in the\nP$\\rm_I$ hierarchy. We propose that our difference equations are discrete\nversions of higher-order Painlev\\\u0027e equations.",
"arxiv_id": "solv-int/9710021",
"authors": [
"Clio Cresswell",
"Nalini Joshi"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "The Discrete Painlev\\\u0027e I Hierarchy",
"url": "https://arxiv.org/abs/solv-int/9710021"
},
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