1996/12/30 by Jarmo Hietarinta, Hietarinta, Jarmo
Mathematics · Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Meromorphic and Entire Functions #nlin.SI #solv-int
paper · pdf · doi:10.48550/arxiv.solv-int/9701002
22 pages, lectures given at the summer school ``The Painleve property, one century later'', Cargese, 3-22 June, 1996
arxiv created 1996/12/30 · openalex publication_date 1996/12/30 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 PVI 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.