Method of Solution for Painleve Equations: Painleve IV, V

摘要

A method to solve the initial value problem associated with the Painleve equations IV, V is given. The procedure involves formulating a Riemann-Hilbert boundary value problem on intersecting lines for the inverse monodromy problem. The boundary value problem is reduced to a sequence of standard problems on single lines in a certain range of parameter squares. Schlesinger transformations allow one to completely cover the parameter space. Special solutions are constructed from special cases of the Riemann problem as well.