Anatolij Prykarpatski

On the Dark Equations, the related integrability theory and applications

Some twenty years ago, a new class of nonlinear dynamical systems, called "dark
equations" was introduced by Boris Kupershmidt, and shown to possess unusual
properties that were not particularly well-understood at that time. Later, in
related developments, some Burgers-type and also Korteweg-de Vries type
dynamical systems were studied in detail, and it was proved that they have in many cases a finite number of conservation laws, a linearization and in some sense degenerate Lax representations, among other properties. In our lectures, we provide a description of a class of self-dual dark-type (or just, dark, for short) nonlinear dynamical systems, which a priori allows their quasi-linearization, whose integrability can be ffectively studied by means of a geometrically motivated gradient-holonomic approach. Moreover, we study a slightly modified form of a self-dual nonlinear dark dynamical system on a functional manifold, whose integrability was recently analyzed by other methods. Not only did this dynamical system appear to be Lax integrable, it also proved to have a rich mathematical architecture including compatible Poisson structures and an infinite hierarchy of nontrivial mutually commuting conservation
laws. We also will demonstrate and prove these properties using the gradient-holonomic integrability scheme devised in our prior work with several collaborators. Finally, we will summarize presented results and indicate some possible future research directions and applications in control theory and other fields of modern mathematical physics.

XXXIX Workshop on Geometric Methods in Physics	19.06–25.06.2022
XI School on Geometry and Physics	27.06–1.07.2022

Event sponsored by:
University of Bialystok