Painleve equations in terms of entire functions

Abstract

In these lectures we discuss how the Painleve equations can be written in terms of entire functions, and then in the Hirota bilinear (or multilinear) form. Hirota's method, which has been so useful in soliton theory, is reviewed and connections from soliton equations to Painleve equations through similarity reductions are discussed from this point of view. In the main part we discuss how singularity structure of the solutions and formal integration of the Painleve equations can be used to find a representation in terms of entire functions. Sometimes the final result is a pair of Hirota bilinear equations, but for $P_{VI}$ we need also a quadrilinear expression. The use of discrete versions of Painleve equations is also discussed briefly. It turns out that with discrete equations one gets better information on the singularities, which can then be represented in terms of functions with a simple zero.