Hamiltonian structure of reductions of the Benney system

Abstract:

The Benney chain (dKP) is a dispersionless integrable system, with a Lie-Poisson Hamiltonian structure. We discuss the Hamiltonian structures of the reductions of this system, in which only finitely many of the variables are independent. These reductions are described by a spectral function - a conformal mapping of the half plane to a slit domain. The reductions are also Hamiltonian,with structures which are found to be of the nonlocal type first described by Ferapontov. We present a procedure relating, in a direct way, the spectral function of a reduction with the metric and the curvature associated with its Poisson structures.