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

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))²u(τ) - u^(τ)τ + 1/τ(-8 ε (u(τ))² + 2ab) + b²u(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.