Spaces of initial values of differential equations with the Painlevé property

Abstract

In this thesis, we study the spaces of initial values of some differential equations with the Painlevé property. The first part of our study begins by introducing the standard techniques associated with such spaces by discussing the well known example of the second Painlevé equation P_II. We then apply these techniques to the cases of linearisable second-order ordinary differential equations (ODEs) and a fourth-order analogue of P_II with particular emphasis on the solutions and structure of the singularities. We explicitly show that the initial value spaces of these ODEs can be regularised for family of general solutions while special family of solutions containing fewer free parameters than the equations’ orders require an infinite number of resolutions or blow ups. To complement our study, we also consider the spaces of initial values of partial differential equations (PDEs). Our examples are Burgers’ and the Korteweg-de Vries equations, whose movable singularities are described by Laurent expansions of the solutions around an arbitrary noncharacteristic manifold. We embed the initial values of these PDEs in complex projective spaces of the appropriate respective dimension and resolve base loci in the corresponding space. As in the ODE case, the initial value space is best understood as a foliation. It is interesting to observe that for both the PDEs, generic initial values are resolved by a finite number of blow ups, while certain initial values lead to an infinite number of blow ups. We provide evidence to show that the latter cases correspond to implicit special solutions. All of the resolutions are described explicitly. Our results suggest that the geometric framework of initial value spaces for the Painlevé equations extends to integrable PDEs. While not all the correspondences between the two frameworks are pursued in this thesis, they suggest tantalising rich directions for future research.