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 τ → ±i∞ for solutions of the degenerate Painlevé III equation (DP3E), 𝑢′′(τ) = (𝑢′(τ))²/𝑢(τ) − 𝑢′(τ)/τ + 1/τ(−8ε(u(τ))² + 2𝑎𝑏) + 𝑏²/𝑢(τ), where ε ∈ {±1}, 𝑎 ∈ ℂ, and 𝑏 ∈ ℝ∖{0}, are parametrised in terms of the monodromy data of an associated first-order 2 × 2 matrix linear ODE 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 various Lie-point symmetries for the DP3E are derived.