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

TL;DR

This paper derives a Kadomtsev-Petviashvili type equation in conformal variables for weakly transversal surface waves over topography, extending conformal mapping techniques to more complex geometries.

Contribution

It introduces a novel conformal variables formulation for weakly transversal waves over topography, accommodating non-smooth topographies via asymptotic analysis.

Findings

The derived equation extends existing weakly nonlinear dispersive wave models.

Numerical simulations validate the applicability of the new model.

The approach handles complex topographies without requiring smoothness.

Abstract

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.