Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation: II

Abstract

The degenerate third Painlevé equation, u ′′ (τ ) = (u ′ , and a ∈ C, is studied via the Isomonodromy Deformation Method. Asymptotics of general regular and singular solutions as τ → ±∞ and τ → ±i∞ are derived and parametrized in terms of the monodromy data of the associated 2 × 2 linear auxiliary problem introduced in [1]. Using these results, three-real-parameter families of solutions that have infinite sequences of zeroes and poles that are asymptotically located along the real and imaginary axes are distinguished: asymptotics of these zeroes and poles are also obtained.

Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin

“…where σ 3 = 1 0 0 −1 and the sectors Ω ∞ k are defined on p.1170 of [7]. The last term, − a 2 ln τ in the exponent above is absent in [7] and [8]. This incorrectness, does not have any effect on the definitions and results presented in Sections 2 and 3 of these papers.…”

“…I also present there a few plots of u(τ ) and its large asymptotics for the pure imaginary and real negative values of the parameter a. While writing Sections 7 and 8 I noticed and corrected a few (nondramatic) faults in our works [7] and [8], this information will be useful for the readers interested in the results concerning any other solutions of equation (1.1).…”

“…Since such tricks might be too involved for the reader who just need to use the result I dedicated Section 7 for the detailed derivation of the monodromy data. In Section 8 I use the monodromy data obtained in Section 7 and the results given in papers [7] and [8] to get asymptotics of u(τ ) for the large values of τ . I also present there a few plots of u(τ ) and its large asymptotics for the pure imaginary and real negative values of the parameter a.…”

Asymptotics of Integrals of Some Functions Related to the Degenerate Third Painlevé Equation

“…For a = 0, the solution holomorphic in a neighbourhood of τ = 0 and vanishing at τ = 0 does not exist. For a > 0, such a solution has an infinite number of poles on the real axis, which can be deduced from the results of [2]. Therefore, only the case a < 0 is considered below.…”

“…Asymptotics of the function u were studied in [1,2]: the corresponding asymptotics for the function ϕ can also be extracted from these papers. In order to do so, recall that in Proposition 1.2 of [1] there was one more function:…”