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The Painlevé‐Kowalevski and Poly‐Painlevé Tests for Integrability
Author(s) -
Kruskal Martin D.,
Clarkson Peter A.
Publication year - 1992
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm199286287
Subject(s) - mathematics , gravitational singularity , point (geometry) , ordinary differential equation , space (punctuation) , mathematical analysis , phase space , singular point of a curve , differential equation , calculus (dental) , geometry , computer science , physics , thermodynamics , operating system , medicine , dentistry
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 Painleve 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.