Euler-Poincaré flows on sl_n Opers and Integrability

Partha Guha

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Submission date: 05. Dec. 2006
Pages: 34
published in: Acta applicandae mathematicae, 95 (2007) 1, p. 1-30 
DOI number (of the published article): 10.1007/s10440-006-9078-6
Bibtex
MSC-Numbers: 53A07, 53B50, 35Q53
Keywords and phrases: opers, Virasoro action, projective structure
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Abstract:
We consider the action of vector field on the space of an - opers on , i.e., a space of nth order differential operator . This action takes the sections of to those of , where is the cotangent bundle on . In this paper we study Euler-Poincaré (EP) flows on the space of opers, In particular, we demonstrate explicitly EP flows on the space of third and fourth order diffrential operators (or and 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 oper defines an immersion in homogeneous coordinates. We derive the Schwarzian KdV equation as an evolution of the solution curve associated to , We study the factorization of higher order operators and its compatibility with the action of . We obtain the generalized Miura transformation and its connection to the modified Boussinesq equation for oper. We also study the eigenvalue problem associated to oper. We discuss flows on the special higher order differential operators for all and its connection to KdV equation. Finally we explore a relation between projective vector field equation and generalized Riccati equations.