Euler-Poincaré Flows on the Loop Bott-Virasoro Group and Space of Tensor Densities and 2 + 1 Dimensional Integrable Systems
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Submission date: 23. Jul. 2009
published in: Reviews in mathematical physics, 22 (2010) 5, p. 485-505
DOI number (of the published article): 10.1142/S0129055X10003989
MSC-Numbers: 53A07, 53B50
Keywords and phrases: loop Virasoro algebra, Calogero-Bogoyavlenskii-Schiff equation, 2+1 -dimensional Camassa equation
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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 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.