Apparent singularities of the finite-depth Zakharov equation

Abstract

The description of weakly nonlinear water-wave evolution over a horizontal bottom by the integro-differential Zakharov equation, because of utilising the underlying Hamiltonian structure, has many advantages over direct use of the Euler equations. However, its application to finite-depth situations is not straightforward since, in contrast to the deep-water case, the kernels governing the four-wave interactions are singular, as well as the kernels in the canonical transformation that removes non-resonant interactions from the original equations of motion. At the singularities, these kernels are finite but not unique. The issue of how to use the Zakharov equation for finite depth and whether it is possible at all was debated intensely in the literature for decades but remains outstanding. Here we show that the absence of a limit of the kernels at the singularities is inconsequential, since in the equations of motion it is only the integral that matters. By applying the definition of the Dirac- $\delta$ , we show that all the integrals involving a trivial manifold singularity are evaluated uniquely. Therefore, the Zakharov evolution equation and the nonlinear canonical transformation are only apparently singular. The findings are validated by application to examples where predictions based on the Zakharov equation are compared with known solutions obtained from the Euler equations.