Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
40/2009

Euler-Poincar\'e Flows on the Loop Bott-Virasoro Group and Space of Tensor Densities and $2+1$ Dimensional Integrable Systems

Partha Guha

Abstract

Following the work of Ovsienko and Roger (Comm. Math. Phys. 273 (2007) 357-378) we study a new kind of deformation of loop Virasoro algebra. Using this new algebra we formulate the Euler-Poincaré flows on the coadjoint orbit of loop Virasoro algebra. We show that the Calogero-Bogoyavlenskii-Schiff equation and various other $(2+1)$-dimensional Korteweg--deVries (KdV) type systems follow from this construction. Using the right invariant $H^1$ inner product on the Lie algebra of loop Bott-Virasoro group we formulate Euler-Poincaré framework of the $2+1$-dimensional of the Camassa-Holm equation. This equation appears to be the Camassa-Holm analogue of the Calogero-Bogoyavlenskii-Schiff type $2 + 1$-dimensional KdV equation. We also derive the $(2+1)$-dimensional generalization of the Hunter-Saxton equation.

Finally, we give an Euler-Poincaré formulation of one-parameter family of $1+1$-dimensional partial differential equations, known as the b-field equations. Later we extend our construction to algebra of loop tensor densities to study the Euler-Poincaré framework of the $2+1$-dimensional extension of b-field equations.

Received:
Jul 23, 2009
Published:
Jul 24, 2009
MSC Codes:
53A07, 53B50
Keywords:
loop Virasoro algebra, Calogero-Bogoyavlenskii-Schiff equation, 2+1 -dimensional Camassa equation

Related publications

inJournal
2010 Repository Open Access
Partha Guha

Euler-Poincaré flows on the loop Bott-Virasoro group and space of tensor densities and (2 + 1)-dimensional integrable systems

In: Reviews in mathematical physics, 22 (2010) 5, pp. 485-505