Hamiltonian description of nonlinear processes in incompressible liquids with a free boundary

We propose the Hamiltonian approach to the hydrodynamics of a compressible liquid with a free boundary based on perturbation theory and analytical continuation. Amplitudes of the bulk and surface waves play the role of canonical variables. We argue that there exist no local canonical variables. For a simplified model of a barotropic liquid we have found the decay amplitude of the bulk sound into two surface waves and the amplitude of the Cherenkov radiation of a surface wave generated by the bulk sound.