One-Parameter Meromorphic Solution of the Degenerate Third Painlevé Equation with Formal Monodromy Parameter a=± i/2 Vanishing at the Origin

2023/05/26 by A. V. Kitaev, Kitaev, A. V., A. Vartanian +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Meromorphic and Entire Functions #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2305.17278

Abstract

We prove that there exists a one-parameter meromorphic solution u(τ) vanishing at τ=0 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 formal monodromy parameter a=± i/2. We study number-theoretic properties of the coefficients of the Taylor-series expansion of u(τ) at τ=0 and its asymptotic behaviour as τ→+∞. These asymptotics are visualized for generic initial data.

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