arXiv subjects

Thomas Kecker

Publications and source records attributed to Thomas Kecker.

6 recordsLinked to original sources

arXiv2026

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

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.

nlin.SI↗

arXiv2024

Spaces of initial conditions for quartic Hamiltonian systems of Painlev\'e and quasi-Painlev\'e type

The geometric approach for Painlev\'e and quasi-Painlev\'e differential equations in the complex plane is applied to non-autonomous Hamiltonian systems, quartic in the dependent variables. By computing their defining manifolds (analogue of the Okamoto's space of initial conditions in the quasi-Painlev\'e case), we provide a classification of such systems. We distinguish the various cases by the local behaviour at the movable singularities of the solutions, which are algebraic poles or ordinary poles. The principal cases are categorised by the initial base points of the system in the extended phase space $\mathbb{CP}^2$ and their multiplicities, arising from the coalescence of $4$ simple base points in the generic case. Through the mechanisms of coalescence of base points and degeneration (by setting certain coefficient functions in the Hamiltonian to $0$), all possible sub-cases of quartic Hamiltonian systems with the quasi-Painlev\'e property are obtained, and are characterised by their corresponding Newton polygons. As particular sub-cases we recover certain systems equivalent to known Painlev\'e equations, or variants thereof. The resulting picture is a multi-faceted description of each case: the local behaviour around singularities, the surface type, and the Newton polygon.

nlin.SI↗

arXiv2024

Geometric approach for the identification of Hamiltonian systems of quasi-Painlev\'e type

Some new Hamiltonian systems of quasi-Painlev\'e type are presented and the analogue of Okamoto's space of initial conditions computed. Using the geometric approach that was introduced originally for the identification problem of Painlev\'e equations, comparing the irreducible components of the inaccessible divisors arising in the blow-up process, we find bi-rational coordinate changes between some of these systems that give rise to the same global Hamiltonian structure. This scheme thus gives a method for identifying Hamiltonian systems up to bi-rational maps, which is performed in this article for systems of quasi-Painlev\'e type having singularities that are either square-root type algebraic poles or ordinary poles.

math.CA↗

arXiv2022

Regularising Transformations for Complex Differential Equations with Movable Algebraic Singularities

In a 1979 paper, K. Okamoto introduced the space of initial values for the six Painlevé equations and their associated Hamiltonian systems, showing that these define regular initial value problems at every point of an augmented phase space, a rational surface with certain exceptional divisors removed. We show that the construction of the space of initial values remains meaningful for certain classes of second-order complex differential equations, and more generally, Hamiltonian systems, where all movable singularities of all their solutions are algebraic poles (by some authors denoted the quasi-Painlevé property), which is a generalisation of the Painlevé property. The difference here is that the initial value problems obtained in the extended phase space become regular only after an additional change of dependent and independent variables. Constructing the analogue of space of initial values for these equations in this way also serves as an algorithm to single out, from a given class of equations or system of equations, those equations which are free from movable logarithmic branch points.

math.CA↗

arXiv2017

Space of initial conditions for a cubic Hamiltonian system

In this paper we perform the analysis that leads to the space of initial conditions for the Hamiltonian system $q' = p^2 + zq + α$, $p' = -q^2 - zp - β$, studied by the author in an earlier article. By compactifying the phase space of the system from $\mathbb{C}^2$ to $\mathbb{CP}^2$ three base points arise in the standard coordinate charts covering the complex projective space. Each of these is removed by a sequence of three blow-ups, a construction to regularise the system at these points. The resulting space, where the exceptional curves introduced after the first and second blow-up are removed, is the so-called Okamoto's space of initial conditions for this system which, at every point, defines a regular initial value problem in some coordinate chart of the space.

math.CA↗

arXiv2013

Polynomial Hamiltonian Systems with Movable Algebraic Singularities

The singularity structure of solutions of a class of Hamiltonian systems of ordinary differential equations in two dependent variables is studied. It is shown that for any solution, all movable singularities, obtained by analytic continuation along a rectifiable curve, are at most algebraic branch points.

math.CA↗