Euler–Poincaré Flows on sl n Opers and Integrability

Abstract

We consider the action of vector field Vect(S 1) on the space of an sl n - opers on S 1, i.e., a space of nth order differential operator $\Delta^{(n)} = \frac{d^n}{dx^n} + u_{n-2}\frac{d^{n-2}}{dx^{n-2}}\break + \cdots + u_1\frac{d}{dx} + u_0$ . This action takes the sections of Ω –(n–1)/2 to those of Ω (n+1)/2, where Ω is the cotangent bundle on S 1. In this paper we study Euler–Poincare (EP) flows on the space of sl n opers, in particular, we demonstrate explicitly EP flows on the space of third and fourth order differential operators (or sl 3 and sl 4 opers) and its relation to Drienfeld–Sokolov, Hirota–Satsuma and other coupled KdV type systems. We also discuss the Boussinesq equation associated with the third order operator. The solutions of the sl n oper defines an immersion ${\bf R} \longrightarrow {\Bbb R}P{\kern1pt}^{n-1}$ in homogeneous coordinates. We derive the Schwarzian KdV equation as an evolution of the solution curve associated to Δ (n), we study the factorization of higher order operators and its compatibility with the action of Vect(S 1). We obtain the generalized Miura transformation and its connection to the modified Boussinesq equation for sl 3 oper. We also study the eigenvalue problem associated to sl 4 oper. We discuss flows on the special higher order differential operators for all u i = f(u,u x ,u xx ⋯) and its connection to KdV equation. Finally we explore a relation between projective vector field equation and generalized Riccati equations.