The Degenerate Third Painleve' Equation: Complete Asymptotic Classification of Solutions in the Neighbourhood of the Regular Singular Point

2025/03/14 by Kitaev, A. V., Vartanian, A. · 1 citation
#33E17 #34M30 #34M35 #34M40 #34M55 #34M56 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.11912

Abstract

We give a classification for the small-τ asymptotic behaviours of solutions to the degenerate third Painlevé equation, u^''(τ) = \frac(u^′(τ))²u(τ) - \fracu^′(τ)τ + \frac1τ(-8 ε (u(τ))² + 2ab ) + \fracb²u(τ), ε=±1, ε b>0, a∈ℂ∖ iℤ, in terms of the monodromy data of a 2×2 matrix linear ODE whose isomonodromy deformations they describe. We also study the complete asymptotic expansions of the solutions.

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• Painlevé Property and Generating Functions for Asymptotics

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