Hamiltonian structures of the Melnikov system and its reductions

Abstract

The bi-Hamiltonian structure of an integrable dynamical system introduced by Melnikov (1986) is studied. This equation arises as a symmetry constraint of the KP hierarchy via squared eigenfunctions and can be understood as a Boussinesq system with a source. The standard linear and quadratic Poisson brackets associated with the space of pseudo-differential symbols are used to derive two compatible Hamiltonian operators. A bi-Hamiltonian formulation for the Drinfeld-Sokolov system is derived via reduction techniques.