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Zbl 0803.35128
Kruskal, Martin D.; Clarkson, Peter A.
The Painlevé-Kowalevski and poly-Painlevé tests for integrability. (English)
[J] Stud. Appl. Math. 86, No.2, 87-165 (1992). ISSN 0022-2526; ISSN 1467-9590/e

Summary: The characteristic feature of the so-called Painlevé test for integrability of an ordinary (or partial) analytic differential equation, as usually carried out, is to determine whether all its solutions are single-valued by local analysis near individual singular points of solutions. This test, interpreted flexibly, has been quite successful in spite of various evident flaws.\par We review the Painlevé test in detail and then propose a more robust and generally more appropriate definition of integrability: a multivalued function is accepted as an integral if its possible values (at any given point in phase space) are not dense. This definition is illustrated and justified by examples, and a widely applicable method (the poly- Painlevé method) of testing for it is presented, based on asymptotic analysis covering several singularities simultaneously.

MSC 2000:: *35Q53 KdV-like equations
35-03 Historical (partial differential equations)
37J35 Completely integrable systems, etc.
37K10 Completely integrable systems etc.

Keywords: Painlevé test; definition of integrability

Cited in: Zbl 1029.35210

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