Seminar talk, 29 March 2017

Speaker: Joseph Krasil'shchik

Title: Nonlocal symmetries of Lax integrable equations: a comparative study

Abstract:
We consider four three-dimensional equations: (1) the rdDym equation $u_{ty}=u_{x}u_{xy}-u_{y}u_{xx}$, (2) the 3D Pavlov equation $u_{yy}=u_{tx}+u_{y}u_{xx}-u_{x}u_{xy}$; (3) the universal hierarchy equation $u_{yy}=u_{t}u_{xy}-u_{y}u_{tx}$, and (4) the modified Veronese web equation $u_{ty}=u_{t}u_{xy}-u_{y}u_{tx}$. For each equation, using the know Lax pairs and expanding the latter in formal series in spectral parameter, we construct two infinite-dimensional differential coverings and give a full description of nonlocal symmetry algebras associated to these coverings. For all the four pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not the same) structures: the are (semi) direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras; all of them contain a component of finite grading. We also discuss actions of recursion operators on shadows of nonlocal symmetries.

A joint work with Hynek Baran, Oleg Morozov, and Petr Vojčák.

References:
arXiv:1611.04938