Numerical solution of the Painlevé IV equation

A numerical method for solving the Cauchy problem for the fourth Painlevé equation is proposed. The difficulty of the problem 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 vanishes. The positions of poles and zeros of the solution 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 in the corresponding point and its neighborhood. Numerical results confirming the efficiency of this method are presented.