Bengt Fornberg
Department of Applied Mathematics
University of Colorado, Boulder
Boulder, USA
The six Painlevé equations P-I to P-VI have been the subject of extensive theoretical investigations for about a century. During the last few decades, their range of applications have increased to the extent that their solutions now are ranked among the key special functions of applied mathematics. These solutions often feature extensive pole fields in the complex plane, posing challenges for most numerical methods. This situation changed just over five years ago, with the development of a numerical 'Pole field Solver' (PFS), able to exploit the Painlevé property for fast and accurate numerical solutions also across dense pole fields. In the particularly important case of solutions that are real-valued along the real axis, the complete solution spaces for the P-I, P-II and P-IV equations have now been exhaustively surveyed. Recently, the PFS has been applied also to the remaining three Painlevé equations, for which the solutions in general no longer are single valued. These calculations show that a variety of phenomena can occur on their different Riemann sheets.
The present work has been carried out in collaborations with André Weideman (University of Stellenbosch), Jonah Reeger (US Air Force Institute of Technology), and Marco Fasondini (University of the Free State).