HAMILTONIAN STRUCTURES OF THE MELNIKOV SYSTEM AND ITS REDUCTIONS

The bi-Hamiltonian structure of an integrable dynamical system introdu ced by Melnikov is studied. This equation arises as a symmetry constra int of the KP hierarchy via squared eigenfunctions and can be understo od as a Boussinesq system with a source. The standard linear and quadr atic 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 vi a reduction techniques.