Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter

2023/04/12 by Kitaev, A. V., Vartanian, A. · 1 citation
#20F55 #33E17 #34M30 #34M35 #34M40 #34M55 #34M56 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2304.05671

Abstract

Various properties of algebroid solutions of 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 bgt;0, for the monodromy parameter a=0 are studied. The paper contains connection results for asymptotics as τ→+0 and as τ→+∞ for a∈ℂ. Using these results, the simplest algebroid solution with asymptotics u(τ)→ cτ^1/3 as τ→0, where c∈ℂ∖\0\, together with its associated integral ∫₀^τ(u(t))^-1 d t, are considered in detail, and their basic asymptotic behaviours are visualized.

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• Painlevé Property and Generating Functions for Asymptotics

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