Hamiltonian models for the propagation of irrotational surface gravity waves over a variable bottom

TL;DR

This paper develops Hamiltonian models for irrotational surface gravity waves over a variable bottom, deriving KdV-type equations with variable coefficients and analyzing soliton solutions numerically.

Contribution

It introduces Hamiltonian formulations incorporating variable bottom effects and derives approximate KdV equations with variable coefficients for such waves.

Findings

Derived Hamiltonian equations using Dirichlet-Neumann operators

Obtained KdV equations with variable coefficients accounting for bottom variation

Numerically studied soliton evolution over variable depth

Abstract

A single incompressible, inviscid, irrotational fluid medium bounded by a free surface and varying bottom is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet-Neumann operators. The equations for the surface waves are presented in Hamiltonian form. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types taking into account the effect of the slowly varying bottom. The arising KdV equation with variable coefficients is studied numerically when the initial condition is in the form of the one soliton solution for the initial depth.