Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III (\(D_7\)) Equation
Robert J. Buckingham and Peter D. Miller
RJB: Department of Mathematical Sciences, University of Cincinnati
PDM: Department of Mathematics, University of Michigan, Ann Arbor
Abstract:
It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem.
In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (\(D_7\)) equation valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside to a form of the Weierstraß equation.