dorsal/arxiv
View SchemaPainleve equations in terms of entire functions
| Authors | Jarmo Hietarinta |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9701002 |
| URL | https://arxiv.org/abs/solv-int/9701002 |
Abstract
In these lectures we discuss how the Painleve equations can be written in terms of entire functions, and then in the Hirota bilinear (or multilinear) form. Hirota's method, which has been so useful in soliton theory, is reviewed and connections from soliton equations to Painleve equations through similarity reductions are discussed from this point of view. In the main part we discuss how singularity structure of the solutions and formal integration of the Painleve equations can be used to find a representation in terms of entire functions. Sometimes the final result is a pair of Hirota bilinear equations, but for $P_{VI}$ we need also a quadrilinear expression. The use of discrete versions of Painleve equations is also discussed briefly. It turns out that with discrete equations one gets better information on the singularities, which can then be represented in terms of functions with a simple zero.
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"abstract": "In these lectures we discuss how the Painleve equations can be written in\nterms of entire functions, and then in the Hirota bilinear (or multilinear)\nform. Hirota\u0027s method, which has been so useful in soliton theory, is reviewed\nand connections from soliton equations to Painleve equations through similarity\nreductions are discussed from this point of view. In the main part we discuss\nhow singularity structure of the solutions and formal integration of the\nPainleve equations can be used to find a representation in terms of entire\nfunctions. Sometimes the final result is a pair of Hirota bilinear equations,\nbut for $P_{VI}$ we need also a quadrilinear expression. The use of discrete\nversions of Painleve equations is also discussed briefly. It turns out that\nwith discrete equations one gets better information on the singularities, which\ncan then be represented in terms of functions with a simple zero.",
"arxiv_id": "solv-int/9701002",
"authors": [
"Jarmo Hietarinta"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "Painleve equations in terms of entire functions",
"url": "https://arxiv.org/abs/solv-int/9701002"
},
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"type": "Model",
"variant": "snapshot-2026-03-01",
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