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

A Compelli; R Ivanov; M Todorov

doi:10.1098/rsta.2017.0091

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

Philos Trans A Math Phys Eng Sci. 2018 Jan 28;376(2111):20170091. doi: 10.1098/rsta.2017.0091.

Authors

A Compelli^{1

2}, R Ivanov^{3

2}, M Todorov⁴

Affiliations

¹ School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland.
² Erwin Schrödinger International Institute for Mathematics and Physics, University of Vienna, 1090 Vienna, Austria.
³ School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland rossen.ivanov@dit.ie.
⁴ Department of Differential Equations, Faculty of Applied Mathematics and Informatics, Technical University of Sofia, 8 Kliment Ohridski Boulevard, 1000 Sofia, Bulgaria.

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 Korteweg-de Vries (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.This article is part of the theme issue 'Nonlinear water waves'.

Keywords: Dirichlet–Neumann operators; Korteweg–de Vries equation; solitons; water waves.

Publication types

Review