SPACE OF HIGHER ORDER DIFFERENTIAL OPERATORS ON S1 AND INTEGRABLE FLOWS

Abstract

We study the action of vector field Vect(S1) on the space of higher order differential operators [Formula: see text], and its connection to projective structure on S1. In particular, we study the Euler–Poincaré flows on the space of third order differential operators and its relation to Drienfeld–Sokolov and coupled KdV type systems. We also discuss the Boussinesq equation associated with the third order operator. We study the factorization of higher order operators and its compatibility with the action of Vect(S1). We obtain the generalized Miura transformation and its connection to the modified Boussinesq equation. Finally, we also discuss flows on the special higher order differential operators for all ui = f(u,ux,uxx…) and its connection to KdV equation.