Painleve-III Monodromy Maps Under the D6 → D8 Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions

The third Painleve equation in its generic form, often referred to as PainleveIII(D6), is given by d(2)u/dx(2) = 1/u (du/dx)(2) - 1/x du/dx + alpha u(2) + beta/x + 4u(3) - 4/u, alpha,beta is an element of C. Starting from a generic initial solution u0(x) corresponding to parameters alpha,beta, denoted as the triple (u(0)(x), alpha,beta), we apply an explicit Backlund transformation to generate a family of solutions (u(n)(x), alpha + 4n, beta + 4n) indexed by n. N. We study the large n behavior of the solutions (u(n)(x), alpha + 4n, beta + 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 u(n)(z/n). Our main result is a proof that the limit of solutions u(n)(z/n) exists and is given by a solution of the degenerate Painleve-III equation, known as Painleve-III(D8), d(2)U/dz(2) 1/U (dU/dz )(2) - 1/z dU/dz + 4U(2) + 4/z. A notable application of our result is to rational solutions of Painlev ' e-III(D-6), which are constructed using the seed solution (1, 4m,-4m) where m is an element of C\ Z+ 1/2 ) 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 Painlev ' e-III, both D-6 and D-8 at z = 0. We also deduce the large n behavior of the Umemura polynomials in a neighborhood of z = 0.