Krzysztof Marciniak
Stäckel systems, soliton hierarchies and Painlevé-type systems
The aim of this lecture series is to present a completely new perspective on the relationships between three major classes of integrable systems: classical separable (in the sense of Hamilton–Jacobi theory) Stäckel systems, soliton hierarchies, and Painlevé‑type systems.
I begin by examining how deformations of Stäckel‑separable systems, constructed via suitable algebras of Killing tensors, give rise to both known and new Painlevé‑type equations. I then demonstrate - using the coupled Korteweg–de Vries hierarchy as a guiding example - how appropriate autonomous constraints reduce a soliton hierarchy to a corresponding Stäckel system, highlighting the way in which the geometric structures inherent in the hierarchy manifest themselves in the resulting Stäckel system. Finally, (this part is not yet published), I discuss how suitable non‑autonomous constraints allow one to reduce a soliton hierarchy to a Painlevé‑type system. Thus, a new and complete picture of relations between these three large classes of integrable systems emerges.
The lectures will be self‑contained and will include all necessary definitions and background information.
