Seminar talk, 28 February 2018

From Geometry of Differential Equations
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Speaker: Oleg Morozov

Title: Multi-dimensional Lax-integrable PDEs: extensions, Bäcklund transformations, and nonlocal conservation laws

The talk will be devoted to some recent results of study of three- and four-dimensional Lax-integrable equations. In the first part of the talk I will consider the quasi-classical self-dual Yang--Mills equation and show that the finite-dimensional part of its local symmetry algebra has a nontrivial second exotic cohomology group. The generating cocycle of this cohomology group defines a differential covering, which in its turn generates an integrable extension of the initial equation. In a particular case this covering is the known covering with the non-removable parameter. Also, I will discuss the integrable hierarchy defined by the equation under the study.

In the second part, which is based on the joint work with A.Lelito, I will consider five three-dimensional PDEs whose coverings include non-removable parameters. In the recent joint work with M.V.Pavlov we showed that all these equations are related by Bäcklund transformations. Z.V.Makridin and M.V.Pavlov found a nonlocal conservation law for one of the equations. In the talk I will present nonlocal conservation laws for the other four equations, prove their nontriviality, ans show that all the nonlocal conservation laws for four equations are generated by the Bäcklund transformations from the local conservation law for the Veronese web equation.