Math Seminar "Mathematical Models and Integration Methods"

Negative symmetries: properties and applications One of the definitions of negative symmetry of an integrable equation is given by the formula u_t=(R-a)^{-1}(0) where R is the recursion operator and a is a parameter. This extension of symmetry algebra is of interest from different points of view: 1) negative symmetry can be interesting as an independent equation; 2) it contains information about the entire integrable hierarchy, since the expansion in parameter a serves as a generating function for higher symmetries; 3) there are applications in the problem of constructing finite-dimensional reductions, especially in combination with classical symmetries (which provides an approach to constructing solutions expressed through higher analogues of Painlevé transcendents); 4) there are connections with other constructions, such as squared eigenfunctions symmetries and Bäcklund transformations. In the talk, we consider examples related to the KdV, Boussinesq and Krichever-Noviko