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 for the resonant flow of a fluid over topography. This resonant flow occurs when, in the weakly nonlinear, long-wave limit, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. In addition to the effect of higher-order terms, the effect of boundary-layer viscosity is also considered. These solutions (with and without viscosity) are compared with recent experimental and numerical results.

Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface

“…letting h → H . In this case the coefficients α, β, α 1 , β 1 , γ 1 , γ 2 reduce to the well-known expressions for surface waves in unstratified water of depth H (see, for instance, Marchant and Smyth (1990) …”

Soliton interaction for the extended Korteweg-de Vries equation

“…The one-soliton solution (A* = 0) can be found directly from results given by Kichenassamy & Olver (1992) or by Marchant & Smyth (1990). Expression (2.4) satisfies the extended KdV equation of (2.2) because the solitons are a long distance apart (hence the interaction terms, such as /i/ 2 , in (2.3), are all negligible).…”

“…Marchant & Smyth (1990) derived the higher-order cnoidal wave solution for (1.1). The solitary-wave limit of this solution (their (2.25)) is the same as (2.6) in the special case of one solitary wave (A 2 = 0).…”

“…If effects of higher order are of interest then retention of terms up to the third order in the (small) wave amplitude leads to the extended KdV equation In the context of surface water waves (1.1) is also useful as a first step in understanding the progression from shallow-water waves to deep-water waves, such as occurs in a surf zone. The extended KdV equation of (1.1) with the coefficients given in (1.2) was derived by Marchant & Smyth (1990) to enable resonant flow of a fluid over topography to be modelled more accurately, and it was also derived by Byatt-Smith (1987a) to examine higher-order solitary-wave interactions. Chow (1989) derived (1.1) for surface waves in shallow water subject to a linear shear flow and obtained the coefficients given in (1.2) in the limit of no-shear.…”

“…However, series solutions for higher-order solitary waves which neglect exponentially small nonlocal tails (such as the tail in (1.6)) are good approximations to the exact solution of the extended KdV equation of (1.1) on all but the longest timescales. Marchant & Smyth (1990) Kodama (1985) includes the elements of transformation (1.7) and a nonlocal term. In this paper, transformation (1.7) is extended in a different manner to that used by Kodama (1985), to allow the extended KdV equation with arbitrary coefficients (1.1) to be transformed, to O(a), to the version of the extended KdV equation which is part of the KdV hierarchy of integrable equations ((1.1) with coefficients given by (1.3)).…”