MiS Preprint Repository

We consider the action of vector field $Vect(S^1)$ on the space of an $s{l}_n$ - opers on $S^1$, i.e., a space of $n$th order differential operator ${\mathrm{\Delta }}^{(n)}=\frac{d^n}{d{x}^n}+u_{n-2}\frac{d^{n-2}}{d{x}^{n-2}}+\cdots +u_1\frac{d}{dx}+u_0$. This action takes the sections of ${\mathrm{\Omega }}^{-(n-1)/2}$ to those of ${\mathrm{\Omega }}^{(n+1)/2}$, where $\mathrm{\Omega }$ is the cotangent bundle on $S^1$.

In this paper we study Euler-Poincar\'e (EP) flows on the space of $s{l}_n$ opers, In particular, we demonstrate explicitly EP flows on the space of third and fourth order diffrential operators (or $s{l}_3$ and $s{l}_4$ 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 $s{l}_n$ oper defines an immersion $\mathbf{R}⟶\mathbb{R}{P}^{n-1}$ in homogeneous coordinates. We derive the Schwarzian KdV equation as an evolution of the solution curve associated to ${\mathrm{\Delta }}^{(n)}$,

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 $s{l}_3$ oper. We also study the eigenvalue problem associated to $s{l}_4$ oper. We discuss flows on the special higher order differential operators for all $u_i=f(u,u_x,u_{xx}\cdots )$ and its connection to KdV equation. Finally we explore a relation between projective vector field equation and generalized Riccati equations.