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Correlation between pole location and asymptotic behavior for Painlevé I solutions
Author(s) -
Costin Ovidiu
Publication year - 1999
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199904)52:4<461::aid-cpa3>3.0.co;2-t
Subject(s) - mathematics , gravitational singularity , series (stratigraphy) , mathematical analysis , position (finance) , asymptotic expansion , monotonic function , bounded function , infinity , boundary (topology) , nonlinear system , convergence (economics) , exponential function , paleontology , physics , finance , quantum mechanics , economics , biology , economic growth
We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show by using asymptotic information that the extension provides a method of finding singularities of solutions of nonlinear differential equations. This transasymptotic matching method is applied to Painlevé's first equation, P1. The solutions of P1 that are bounded in some direction towards infinity can be expressed as series of functions obtained by generalized Borel summation of formal transseries solutions; the series converge in a neighborhood of infinity. We prove (under certain restrictions) that the boundary of the region of convergence contains actual poles of the associated solution. As a consequence, the position of these exterior poles is derived from asymptotic data. In particular, we prove that the location of the outermost pole x p ( C ) on ℝ + of a solution is monotonic in a parameter C describing its asymptotics on anti‐Stokes lines and obtain rigorous bounds for x p ( C ). We also derive the behavior of x p ( C ) for large C ∈ ℂ. The appendix gives a detailed classical proof that the only singularities of solutions of P1 are poles. © 1999 John Wiley & Sons, Inc.