The Kadomtsev-Petviashvili equation in conformal variables for waves over topography

The conformal mapping approach is a well established technique for solving the Euler equations for potential flows with one spatial dimension. In this work, we extend this framework to problems with a weakly transversal dependence and, by means of asymptotic expansions, obtain a Kadomtsev-Petviashvili type equation formulated in conformal variables as a model for weakly transversal surface waves propagating over topography. A key advantage of this formulation is that the topography, defined in the physical domain, does not need to be a smooth function, or even a function in the classical sense because, our asymptotic analysis relies on the effective depth, which comes through the Jacobian of the conformal map which is assumed to be a slowly varying function. The resulting equation provides a consistent extension of several well known weakly nonlinear dispersive wave models previously reported in the literature. Numerical simulations are performed to illustrate the newly derived equation.