Dept Math & Stat UiT, Forskningsparken seminar room B459
  5 Mar 2020 Thu 13:00-14:30
  9 Mar 2020 Mon 13:00-14:30
12 Mar 2020 Thu 13:00-14:30

Oleg Morozov (AGH University of Science and Technology in Kraków, Poland)

Integrable systems and extensions of symmetry algebras.

Lax representations, also known as zero-curvature representations, Wahlquist-Estabrook prolongation structures, inverse scattering transformations, or differential coverings, are a key feature of integrable partial differential equations. A number of important techniques for studying integrable PDEs such as Bäcklund transformations, Darboux transformations, recursion operators, and nonlocal symmetries, are based on Lax representations. The challenging unsolved problem in this theory is to find conditions that are formulated in inherent terms of a PDE under study and ensure existence of a Lax representation. The lectures will discuss a recent approach to this problem. We will show that for some PDEs their Lax representations can be inferred from the second exotic cohomology group of the contact symmetry algebras.

The lectures will cover the following topics:

Lecture 1: Introduction. Integrable partial differential equations, Lax representations, differential coverings, Wahlquist-Estabrook forms. Lie algebras, definitions and examples. Symmetry algebras of differential equations.
Lecture 2: Maurer-Cartan forms and structure equations of Lie algebras. Extensions of Lie algebras. Cohomology groups of Lie algebras.
Lecture 3: Extensions of symmetry algebras and Lax representations of integrable systems. Examples.