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MiS Preprint
144/2006

Euler-Poincaré flows on $sl_n$ Opers and  Integrability

Partha Guha

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 $n$th order differential operator $\Delta^{(n)} = \frac{d^n}{dx^n} + u_{n-2}\frac{d^{n-2}}{dx^{n-2}} + \cdots + u_1\frac{d}{dx} + u_0$. This action takes the sections of $\Omega^{-(n-1)/2}$ to those of $\Omega^{(n+1)/2}$, where $\Omega$ is the cotangent bundle on $S^1$.

In this paper we study Euler-Poincar\'e (EP) flows on the space of $sl_n$ opers, In particular, we demonstrate explicitly EP flows on the space of third and fourth order diffrential 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 ${\bf R} \longrightarrow {\Bbb R}P^{n-1}$ in homogeneous coordinates. We derive the Schwarzian KdV equation as an evolution of the solution curve associated to $\Delta^{(n)}$,

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 $sl_3$ oper. We also study the eigenvalue problem associated to $sl_4$ oper. We discuss flows on the special higher order differential operators for all $u_i = f(u,u_x,u_{xx}\cdots)$ and its connection to KdV equation. Finally we explore a relation between projective vector field equation and generalized Riccati equations.

Received:
Dec 5, 2006
Published:
Dec 5, 2006
MSC Codes:
53A07, 53B50, 35Q53
Keywords:
opers, Virasoro action, projective structure

Related publications

inJournal
2007 Repository Open Access
Partha Guha

Euler-Poincaré flows on sln opers and integrability

In: Acta applicandae mathematicae, 95 (2007) 1, pp. 1-30