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The third Painlevé equation in its generic form, often referred to as Painlevé-III (D₆), is given by d²u{ dx²} =1u (du{ dx}) ²-1x du{ dx} + u² + x+4u³-4u, , C. Starting from a generic initial solution u₀ (x) corresponding to parameters, , denoted as the triple (u₀ (x), , ), we apply an explicit Bäcklund transformation to generate a family of solutions (uₙ (x), + 4n, + 4n) indexed by n N. We study the large n behavior of the solutions (uₙ (x), + 4n, + 4n) under the scaling x = z/n in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution uₙ (z/n). Our main result is a proof that the limit of solutions uₙ (z/n) exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III (D₈), d²U{ dz²} =1U (dU{ dz}) ²-1z dU{ dz} + 4U² + 4z. A notable application of our result is to rational solutions of Painlevé-III (D₆), which are constructed using the seed solution (1, 4m, -4m) where m C (Z + 12) and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at z = 0 when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both D₆ and D₈ at z = 0. We also deduce the large n behavior of the Umemura polynomials in a neighborhood of z = 0.
Barhoumi et al. (Sat,) studied this question.