Different Hamiltonians for differential Painlevé equations and their identification using a geometric approach

It is well-known that differential Painlev & eacute; 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 & eacute; equation. Recognizing a Painlev & eacute; equation, for example when it appears in some applied problem, is known as the Painlev & eacute; equivalence problem. Here we consider its Hamiltonian version. We describe a systematic procedure for finding changes of coordinates that transform different Hamiltonian representations of a Painlev & eacute; equation into some canonical form. Our approach is based on Sakai's geometric theory of Painlev & eacute; equations. We explain it in detail for Painlev & eacute; IV, and give a brief summary for Painlev & eacute; V and VI. Such explicit identifications can be helpful for applications, since it gives access to known results about Painlev & eacute; equations, such as structure of symmetries and special solutions for certain parameter values. (c) 2024 Elsevier Inc. All rights reserved.