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MiS Preprint

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

Partha Guha


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.

MSC Codes:
53A07, 53B50
loop Virasoro algebra, Calogero-Bogoyavlenskii-Schiff equation, 2+1 -dimensional Camassa equation

Related publications

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