On Pole-free Solutions to the Painleve Hierarchies

Description

The six Painlevéequations are a series of nonlinear second order differential equations with special properties. Nowadays, it has been realized that they play an important role in many branches of physics, for example, statistical mechanics, nonlinear waves, quantum field theory, general relativity, etc.. Moreover, they may be thought of nonlinear analogues of the classical special functions, such as Airy functions, Bessel functions and parabolic cylinder functions. In the meantime, there has been a considerable amount of interest in their higher order analogues, so-called the Painlevéhierarchies. Usually, the solutions to the Painlevéequations, as well as the Painlevéhierarchies, are meromorphic and have poles in the complex plane. However, there also exist some solutions, which are pole-free on certain line or in certain domain in the complex plane. Although breakthroughs have been made in the recent years, there are still many important open problems in this area.In this project, we will focus on pole-free solutions to the Painlevéhierarchies. More precisely, we will try to prove the existence of these solutions and derive their asymptotics.