The Painleve Equations: A Study on Asymptotic Behaviours and Pole Distribution of their Solutions

Description

The six Painleve equations are a series of second-order nonlinear ordinary differentialequations which were first discovered and studied by Painleve, Gambier and theircolleagues more than one century ago. Although these equations are nonlinear, theyhave very close relations with some linear differential systems. As a consequence, thesolutions of these equations possess fascinating properties which are very different fromthe usual nonlinear ones. For example, when the parameters in the equations equalcertain special values, there are hierarchies of rational solutions or one-parameterfamilies of solutions expressible in terms of the classical special functions, such as Airyfunctions, Bessel functions and parabolic cylinder functions. Nowadays, due to theirimportant applications in many physical branches including statistical mechanics,random matrix theory, nonlinear waves, quantum field theory and general relativity,these solutions are regarded as nonlinear special functions.To understand the Painleve equations in more detail, we will study properties of theirsolutions in this project. More precisely, we will derive the asymptotic expansions of thesolutions on the real axis. Then, the connection formula, which is the relation betweendifferent asymptotic expansions of the same solution, will be established. We will alsoconsider the behavior of the solutions in the complex plane and try to understand howtheir poles are distributed.