Hirota bilinear equations for Painlev\'e transcendents
Classical Analysis and ODEs
2019-05-07 v2 Exactly Solvable and Integrable Systems
Abstract
We present some observations on the tau-function for the fourth Painlev\'e equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary movable zero. The corresponding Taylor series for the tau-functions of the first and second Painlev\'e equations, as well as that for the Weierstrass sigma function, arise naturally as special cases, by setting certain parameters to zero.
Cite
@article{arxiv.1706.02341,
title = {Hirota bilinear equations for Painlev\'e transcendents},
author = {A. N. W. Hone and F. Zullo},
journal= {arXiv preprint arXiv:1706.02341},
year = {2019}
}
Comments
This is version n.2, some typos have been fixed (e.g. formula 31), a formula for the coefficients \hat{A}_j in section 4 has been added, the references have been updated