dorsal/arxiv
View SchemaAnalytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property
| Authors | Martin D. Kruskal, Nalini Joshi, Rod Halburd |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9710023 |
| URL | https://arxiv.org/abs/solv-int/9710023 |
| DOI | 10.1007/BFb0113696 |
Abstract
The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable equations have the Painlev\'e property, that is, all solutions are single-valued around all movable singularities. In this expository article, we review methods for analysing such singularity structure. In particular, we describe well known techniques of nonlinear regular-singular-type analysis, i.e. the Painlev\'e tests for ordinary and partial differential equations. Then we discuss methods of obtaining sufficiency conditions for the Painlev\'e property. Recently, extensions of \textit{irregular} singularity analysis to nonlinear equations have been achieved. Also, new asymptotic limits of differential equations preserving the Painlev\'e property have been found. We discuss these also.
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"abstract": "The integrability (solvability via an associated single-valued linear\nproblem) of a differential equation is closely related to the singularity\nstructure of its solutions. In particular, there is strong evidence that all\nintegrable equations have the Painlev\\\u0027e property, that is, all solutions are\nsingle-valued around all movable singularities. In this expository article, we\nreview methods for analysing such singularity structure. In particular, we\ndescribe well known techniques of nonlinear regular-singular-type analysis,\ni.e. the Painlev\\\u0027e tests for ordinary and partial differential equations. Then\nwe discuss methods of obtaining sufficiency conditions for the Painlev\\\u0027e\nproperty. Recently, extensions of \\textit{irregular} singularity analysis to\nnonlinear equations have been achieved. Also, new asymptotic limits of\ndifferential equations preserving the Painlev\\\u0027e property have been found. We\ndiscuss these also.",
"arxiv_id": "solv-int/9710023",
"authors": [
"Martin D. Kruskal",
"Nalini Joshi",
"Rod Halburd"
],
"categories": [
"solv-int",
"nlin.SI"
],
"doi": "10.1007/BFb0113696",
"title": "Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\\\u0027e Property",
"url": "https://arxiv.org/abs/solv-int/9710023"
},
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