International Laboratory of Cluster Geometry Seminar "Characteristic classes and intersection theory". Speaker : Irina Bobrova
On the Painlev\’e equations and their non-Abelian generalisations
The differential Painlev\’e equations were discovered more than a hundred years ago and since the eighties have been appearing in many branches of mathematics and physics. General solutions of such equations do not have movable singular points (the Painlev\’e property). The Painlev\’e property is closely related to the integrability and that is the reason for the considerable attention to these equations in recent years. During this talk we will focus mostly on the second and fourth Painlev\’e equations.
In the first part of the talk we will give a brief introduction to the Painlev\’e equations and their structures, such as integrability, confluences, monodromy surfaces, Hamiltonians and symmetries. One of the natural generalisations of the Painlev\’e equations is their higher analogs (in other words, hierarchies). It turns out that the hierarchies can inherit some structures of the classical Painlev\’e equations (see, e.g. arXiv:nlin/0610066, arXiv:2010.10617, arXiv:2012.11010).
The second part is devoted to some integrable non-commutative versions of the Painlev\’e equations. In such generalisations, the dependent variables belong to an associative unital algebra, the trivial central extension of which may be given by constants only (in this case, the independent variable is non-commutative) or by the independent variable. We will consider examples of both the generations mentioned above and a method called the matrix Painlev\’e-Kovalevskaya test that allows us to detect integrable matrix generalisations of the Painlev\’e equations. This part is based on a joint work with Vladimir Sokolov (arXiv: 2107.11680, arXiv:2110.12159).