Seminar talk, 29 March 2017

From Geometry of Differential Equations
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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 [math]\displaystyle{ u_{ty} = u_x u_{xy} - u_y u_{xx} }[/math], (2) the 3D Pavlov equation [math]\displaystyle{ u_{yy} = u_{tx} + u_y u_{xx} - u_x u_{xy} }[/math]; (3) the universal hierarchy equation [math]\displaystyle{ u_{yy} = u_t u_{xy} - u_y u_{tx} }[/math], and (4) the modified Veronese web equation [math]\displaystyle{ u_{ty} = u_t u_{xy} - u_y u_{tx} }[/math]. 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