Trans-Series Asymptotics of Solutions to the Degenerate Painlevé III Equation: A Case Study

Abstract

A one-parameter family of trans-series asymptotics as τ → ±∞ and as τ → ±i∞ for solutions of the degenerate Painlevé III equation (DP3E),, where ε ∈ {±1}, a ∈ C, and b ∈ R\{0}, are parametrised in terms of the monodromy data of an associated 2×2 linear auxiliary problem via the isomonodromy deformation approach: trans-series asymptotics for the associated Hamiltonian and principal auxiliary functions and the solution of one of the σ-forms of the DP3E are also obtained. The actions of Lie-point symmetries for the DP3E are derived.

Meromorphy of solutions for a wide class of ordinary differential equations of Painlevé type

“…which coincides with the system of ODE ( 69), (70) in [54, §5.1]. The component u of a solution of this system satisfies the ODE…”

Boutroux ansatz for the degenerate third Painlevé transcendents

“…As mentioned in [17], [27], in physical and geometrical applications, degenerate P III appears in contexts independent of complete P III , and its significant analytic properties are important. Indeed behaviours of solutions of (1.1) along real and imaginary axes [17], [18] are quite different from those for complete P III [12].…”

“…Using the isomonodromic property of (3.1), Kitaev and Vartanian [17], [18] obtained asymptotic solutions of (1.1) as τ → ±∞, ±i∞ and τ → ±0, ±i0, with connection formulas among them. Furthermore, for (1.1), a special meromorphic solution is studied by [16], [19], and truncated solutions are given by [27].…”