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The Painlevé-Kowalevski and poly-Painlevé tests for integrability. (English) Zbl 0803.35128

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.

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:
35Q53KdV-like (Korteweg-de Vries) equations
35-03Historical (partial differential equations)
37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies