Preprint 144/2006

Euler-Poincaré flows on sln 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
MSC-Numbers: 53A07, 53B50, 35Q53
Keywords and phrases: opers, Virasoro action, projective structure
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We consider the action of vector field formula24 on the space of an formula26 - opers on formula28, i.e., a space of nth order differential operator formula32. This action takes the sections of formula34 to those of formula36, where formula38 is the cotangent bundle on formula28. In this paper we study Euler-Poincaré (EP) flows on the space of formula26 opers, In particular, we demonstrate explicitly EP flows on the space of third and fourth order diffrential operators (or formula44 and formula46 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 formula26 oper defines an immersion formula50 in homogeneous coordinates. We derive the Schwarzian KdV equation as an evolution of the solution curve associated to formula52, We study the factorization of higher order operators and its compatibility with the action of formula24. We obtain the generalized Miura transformation and its connection to the modified Boussinesq equation for formula44 oper. We also study the eigenvalue problem associated to formula46 oper. We discuss flows on the special higher order differential operators for all formula60 and its connection to KdV equation. Finally we explore a relation between projective vector field equation and generalized Riccati equations.

19.04.2013, 01:41