TY - JOUR

T1 - Euler-Poincaré flows on sl n opers and integrability

AU - Guha, Partha

N1 - Funding Information:
Acknowledgements It is my pleasant duty to acknowledge gratefully for several stimulating discussions with Professors Sasha Kirillov, Jerry Marsden, Tudor Ratiu, Jose Carinena, George Wilson and Valentin Ovsienko. I am particularly grateful to Professor Valentin Ovsienko for introducing me the subject and to Professor Jerry Marsden for urging me to study EP flows. I am thankful to Professor David Ben-Zvi for explanation of projective connections and opers on S1. Finally I would like to thank Professor Dieter Mayer at TU Clausthal and Professor Jürgen Jost at the Max Planck Insitute for Mathematics in the Sciences for their gracious hospitality where the paper was finished for kind hospitality and stimulating atmosphere. This work has been partially supported by the DFG Research Group “Zeta functions and locally symmetric spaces” which is gratefully acknowledged.

PY - 2007/1

Y1 - 2007/1

N2 - 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 Δ(n)=dn/dxn+u n-2 dn-2/dxn-2+⋯+u1d/ dx+u0. 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-Poincaré (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 ℝ→ℝPn-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,ux,u xx ⋯) and its connection to KdV equation. Finally we explore a relation between projective vector field equation and generalized Riccati equations.

AB - 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 Δ(n)=dn/dxn+u n-2 dn-2/dxn-2+⋯+u1d/ dx+u0. 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-Poincaré (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 ℝ→ℝPn-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,ux,u xx ⋯) and its connection to KdV equation. Finally we explore a relation between projective vector field equation and generalized Riccati equations.

KW - Coupled KdV equation Boussinesq equation and Riccati

KW - Drienfeld-Sokolov equation

KW - Hirota-Satsuma equation

KW - Opers

KW - Projective structure

KW - Virasoro action

UR - http://www.scopus.com/inward/record.url?scp=33847011361&partnerID=8YFLogxK

U2 - 10.1007/s10440-006-9078-6

DO - 10.1007/s10440-006-9078-6

M3 - Article

AN - SCOPUS:33847011361

SN - 0167-8019

VL - 95

SP - 1

EP - 30

JO - Acta Applicandae Mathematicae

JF - Acta Applicandae Mathematicae

IS - 1

ER -