Nonisospectral symmetries of the KdV equation and the corresponding symmetries of the Whitham equations

Abstract

In our paper we construct a new infinite family of symmetries of the Whitham equations (averaged Korteveg-de-Vries equation). In contrast with the ordinary hydrodynamic-type flows these symmetries are nonhomogeneous (i.e. they act nontrivially at the constant solutions), are nonlocal, explicitly depend upon space and time coordinates and form a noncommutative algebra, isomorphic to the algebra of the polynomial vector fields in the complex plane (Virasoro algebra with the zero central charge).