A Hamiltonian Approach to Nonlinear Modulation of Surface Water Waves

Many wave phenomena in physics are described by weakly nonlinear nearly monochromatic solutions in the form of modulated wave packets. The examples include ocean waves as well as waves in optics and in plasmas. There are a number of approaches to deriving the envelope equations for these theories of amplitude modulation. In this paper, we give a unified approach, based on the principles of a Hamiltonian formulation of the equations of motion. Our principal example is the system of equations of free surface water waves, for which we give a derivation of the classical nonlinear Schrödinger and Davey–Stewartson equations, as well as the higher-order Dysthe system. One consequence of our analysis from this point of view is that the Dysthe equation can be posed as a Hamiltonian PDE.