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Euler-Poincare formalism of coupled KdV type system and diffeomorphism group on S1

Guha P.

Journal of Applied Analysis

2005

Vol. 11, nr 2

261-282

This paper describes a wide class of coupled KdV equations. The first set of equations directly follow from the geodesic flows on the Bott-Virasoro group with a complex field. But the set of 2-component systems of nonlinear evolution equations, which includes dispersive water waves, Ito's equation, many other known and unknown equations, follow from the geodesic flows of the right invariant L2 metric on the semidirect product group [wzór], where Diff(S1) is the group of orientation preserving diffeomorphisms on a circle. We compute the Lie-Poisson brackets of the Antonowicz-Fordy system, and the mode expansion of these beackets yield the twisted Heisenberg-Virasoro algebra. We also give an outline to study geodesic flows of a H1 metric on [wzór].