Numerical solution of the Painlevé VI equation

A numerical method for solving the Cauchy problem for the sixth Painlevé equation is proposed. The difficulty of this problem, as well as the other Painlevé equations, is that the unknown function can have movable singular points of the pole type; moreover, the equation may have singularities at the points where the solution takes the values 0 or 1 or is equal to the independent variable. The positions of all of these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. The main results of this paper are the derivation of the auxiliary equations and the formulation of transition criteria. Numerical results illustrating the potentials of this method are presented.