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The Coalescence Limit of the Second Painlevé Equation
Author(s) -
Halburd Rod,
Joshi Nalini
Publication year - 1996
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/sapm19969711
Subject(s) - gravitational singularity , limit (mathematics) , mathematics , coalescence (physics) , mathematical physics , mathematical analysis , limit point , physics , astrobiology
In this paper, we study a well‐known asymptotic limit in which the second Painlevé equation (P II ) becomes the first Painlevé equation (P I ). The limit preserves the Painlevé property (i.e., that all movable singularities of all solutions are poles). Indeed it has been commonly accepted that the movable simple poles of opposite residue of the generic solution of P II must coalesce in the limit to become movable double poles of the solutions of P I , even though the limit naively carried out on the Laurent expansion of any solution of P II makes no sense. Here we show rigorously that a coalescence of poles occurs. Moreover we show that locally all analytic solutions of P I arise as limits of solutions of P II .