Systematic construction of nonautonomous Hamiltonian equations of Painlevé type. I. Frobenius integrability

Abstract

This article is the first one in a suite of three articles exploring connections between dynamical systems of Stäckel type and of Painlevé type. In this article, we present a deformation of autonomous Stäckel-type systems to nonautonomous Frobenius integrable systems.First, we consider quasi-Stäckel systems with quadratic in momenta Hamiltonians containing separable potentials with time-dependent coefficients, and then, we present a procedure of deforming these equations to nonautonomous Frobenius integrable systems. Then, we present a procedure of deforming quasi-Stäckel systems with so-called magnetic separable potentials to nonautonomous Frobenius integrable systems. We also provide a complete list of all two-and three-dimensional Frobenius integrable systems, both with ordinary and with magnetic potentials, which originate in our construction. Further, we prove the equivalence between both classes of systems. Finally, we show how Painlevé equations 𝑃 𝐼 − 𝑃 𝐼𝑉 can be derived from our scheme.

Systematic construction of non-autonomous Hamiltonian equations of Painlevé-type. III. Quantization

“…In the second article (Part II, i.e. [10]) we have constructed the isomonodromic Lax representations for these systems, proving that Frobenius integrable systems constructed in Part I are indeed Painlevé-type systems. Each of such families was written in two different representations, called an ordinary one and a magnetic one, respectively, and they were connected by a multi-time canonical transformation [14].…”

“…In the first paper (Part I, i.e. [9], see also [8]) we have constructed, starting from appropriate Stäckel-type systems, multi-parameter families of Frobenius integrable non-autonomous Hamiltonian systems with arbitrary number of degrees of freedom. In the second article (Part II, i.e.…”

“…The scalar potentials V (α) r can be explicitly constructed by a recursion formula [5], [9]. Finally, the basic separable vector potentials P (γ) r in Viète coordinates have the form…”

Systematic construction of nonautonomous Hamiltonian equations of Painlevé‐type. II. Isomonodromic Lax representation

“…In case that $$\varphi$$ is a Laurent sum $$\varphi (x)=\sum _{\gamma }\varepsilon _{\gamma }x^{\gamma }$$, $$M_{r}^{\varphi }$$ will be the corresponding Laurent sum $$M_{r}^{\varphi }(\lambda ,\mu )=\sum _{\gamma }\varepsilon _{\gamma }M_{r} ^{(\gamma )}$$ of basic separable magnetic potentials $$M_{r}^{(\gamma )}$$. They have the explicit form $$\begin{equation} M_{r}^{(\gamma )}=\sum _{i=1}^{n}\frac{\partial \rho _{r}}{\partial \lambda _{i} }\frac{\lambda _{i}^{\gamma }\mu _{i}}{\Delta _{i}},\quad \Delta _{i}=\prod \limits _{j\ne i}(\lambda _{i}-\lambda _{j}) \end{equation}$$(see Part I 1 for more details) and are called magnetic because they depend linearly on momenta $$\mu _{i}$$.…”

“…This is the second article in the suite of articles investigating a systematic way of constructing Painlevé‐type systems from an appropriate Stäckel‐type systems. In the previous paper (Part I 1 ) we have constructed multiparameter families of Frobenius integrable nonautonomous Hamiltonian systems with arbitrary number of degrees of freedom. Each of these families was written in two different representations (two different coordinate systems) that we called an ordinary one and a magnetic one, respectively, connected by a multitime‐dependent canonical transformation 2…”