Different Hamiltonians for differential Painlev\'e equations and their identification using a geometric approach

TL;DR

This paper presents a geometric method to identify and transform different Hamiltonian forms of Painlevé equations into canonical forms, aiding in recognizing equations and linking various problems with the same structure.

Contribution

It introduces a systematic geometric approach based on Sakai's theory to find coordinate transformations between Hamiltonian representations of Painlevé equations.

Findings

Detailed procedure for P_IV Hamiltonian transformations

Application to P_V and P_VI cases

Potential for adaptation to other Painlevé equations

Abstract

It is well-known that differential Painlev\'e equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique -- there are many very different Hamiltonians that result in the same differential Painlev\'e equation. Recognizing a Painlev\'e equation, for example when it appears in some applied problem, is known as the \emph{Painlev\'e equivalence problem}, and the question that we consider here is the Hamiltonian form of this problem. Making such identification explicit, on the level of coordinate transformations, can be very helpful for an applied problem, since it gives access to the wealth of known results about Painlev\'e equations, such as the structure of the symmetry group, special solutions for special values of the parameters, and so on. It can also provide an explicit link between different problems that have the same underlying structure. In this paper we describe a systematic procedure for finding changes of coordinates that trasform different Hamiltonian representations of a Painlev\'e equation into some chosen canonical form. Our approach is based on Sakai's geometric theory of Painlev\'e equations. We explain this procedure in detail for the fourth differential ${\text{P}_{\mathrm{IV}}}$ equation, and also give a brief summary of some known examples for ${\text{P}_{\mathrm{V}}}$ and ${\text{P}_{\mathrm{VI}}}$ cases. It is clear that this approach can easily be adapted to other examples as well, so we expect our paper to be a useful reference for some of the realizations of Okamoto spaces of initial conditions for Painlev\'e equations.