The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography

Abstract

The extended Korteweg-de Vries equation which includes nonlinear and dispersive terms cubic in the wave amplitude is derived from the water-wave equations and the Lagrangian for the water-wave equations. For the special case in which only the higher-order nonlinear term is retained, the extended Korteweg-de Vries equation is transformed into the Korteweg-de Vries equation. Modulation equations for this equation are then derived from the modulation equations for the Korteweg-de Vries equation and the undular bore solution of the extended Korteweg-de Vries equation is found as a simple wave solution of these modulation equations. The modulation equations are also used to extend the solution fo..