MiS Preprint Repository

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.