On the non-uniqueness of the kernel of the Zakharov equation in intermediate and shallow water: the connection with the Davey-Stewartson equation

The Zakharov equation describes the evolution of weakly nonlinear surface gravity waves for arbitrary spectral shape. For deep-water waves, results from the Zakharov equation are well established. However, for two-dimensional propagation, in intermediate and shallow water, there are problems related to the treatment of apparent singularities in the contribution of the wave-induced set-up to the evolution of the surface gravity waves. More specifically, the kernel in the integral term is characterized by a regular and an apparent singular contribution. Here, we show that the Davey-Stewartson equation can be directly derived from the Zakharov equation, also in the shallow water limit. This result provides guidance on how to treat the singular contribution to the evolution of the action variable. A relevant result that is obtained is that the growth rate obtained from the stability analysis of a plane wave in shallow water does not depend on the singular part of the kernel of the Zakharov equation.