Modified Painlev\'e systems with meromorphic solutions for polynomial Hamiltonians of all degrees

TL;DR

This paper classifies polynomial Hamiltonian systems with all solutions meromorphic in the complex plane, extending Painlevé systems and identifying new Hamiltonians for various Painlevé equations.

Contribution

It provides a complete classification of polynomial Hamiltonians with meromorphic solutions for degrees three, four, five, and seven, including new Hamiltonians not previously documented.

Findings

Classified all such systems up to degree seven.

Identified 12 standard Hamiltonians serving as references.

Discovered new Hamiltonians for Painlevé I, II, III, IV, V, and VI.

Abstract

We review non-autonomous Hamiltonian systems, polynomial in two dependent variables, with the property that all of their solutions are meromorphic functions in the complex plane. These are related to known Hamiltonian systems with the Painlev\'e property, for which the solutions are single-valued outside a set of fixed singularities. Our systems are equivalent to them in the absence of fixed singularities, and give modified Painlev\'e equations otherwise. Using the geometric approach by computing the Okamoto's spaces of initial conditions for certain Hamiltonian systems with general coefficient functions, we obtain differential constraints on these functions for the systems to have only meromorphic solutions. Guided by the Newton polygon of the Hamiltonian function, we obtain all such systems with polynomial Hamiltonian of degree three, four, five, and seven, up to affine equivalence in the dependent variables, while there are none for degree six or degree higher than seven. We thus obtain a list of 12 standard polynomial Hamiltonians that can serve as reference for the Painlev\'e equivalence problem. This list contains also some new Hamiltonians not previously written down, such as quartic Hamiltonians for Painlev\'e I and II, quartic Hamiltonians for the modified Painlev\'e III and V equations, a quintic Hamiltonian for Painlev\'e IV and quintic and septic Hamiltonians for a modified Painlev\'e VI equation.