Negative symmetries: properties and applications

Abstract: 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-Novikov equations and the Volterra lattice.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods