Isomonodromic deformation methods for the initial value problems of the painleve equations

Abstract

The Painleve equations appeared in the theory of the ordinary differential equations at the beginning of our century, in connection with a classification problem for the equations of the form: q"=F(q',q,x) where F is rational in q', algebraic in q and locally analytic in x, whose solutions are free from movable critical points. This dissertation deals with the initial value problems of some Painleve equations via monodromy preserving deformation methods. The dissertation is divided into three chapters. Chapter 1 presents preliminary introduction and information about the forthcoming chapters. General theory about Painleve equations is discussed. A brief summary about the solutions of matrix linear differ ential equations, monodromy preserving deformation theory and the Stokes phenomenon is given here. In chapter 2 and 3, respectively the initial value problems of 2nd and 4th Painlevâ equations are solved via isomonodromic deformation theory, Special solutions of these Painleve equatios expressible in terms of the classical transcendental functions are obtained.