A second-order theory for solitary waves in shallow fluids

Abstract

Solitary waves in density stratified fluids of shallow depth are described, to first order in wave amplitude, by the Korteweg–de Vries equation; the solution for a single solitary wave has the familiar ‘‘sech2’’ profile and a phase speed which varies linearly with the wave amplitude. This theory is here extended to second order in wave amplitude. The second-order correction to the wave profile and the phase speed and the first-order correction to the wavelength are all determined. Four special cases are discussed in detail. In certain special circumstances the first-order theory may fail due to the vanishing of the nonlinear coefficient in the Korteweg–de Vries equation. When this occurs a different theory is required which leads to an equation with both quadratic and cubic nonlinearities.

Analysis of internal solitary waves observed in Davis Strait

“…Relation between half-amplitude width and displacement using the eigenfunctions obtained from: (a) CTD 1-16 without mean flow and (b) CTD 1-16 with Flow 2. The dashed line gives this relation for the second-order KdV theory of Gear and Grimshaw (1983). Finally in Fig.…”

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

“…A more detailed analysis of the properties of the steady-state solitary waves in a fluid with arbitrary density and current stratification, valid to the second order of an asymptotic expansion was reported by Gear and Grimshaw (1983). They calculated the shape of the internal solitary wave and its speed for different models of the fluid stratification.…”

Soliton interaction for the extended Korteweg-de Vries equation

“…In addition, (1.1) describes internal waves of moderate amplitude in a shallow, densitystratified fluid. Gear & Grimshaw (1983) calculated second-order solitary-wave solutions for various density stratifications and shear flows; in the limit of no-shear and no stratification (1.2) would also be obtained. The question arises as to whether the higher-order solitary-wave solutions of (1.1) are solitons or not, that is, whether or not they undergo elastic collisions.…”