A second-order theory for solitary waves in shallow fluids
- 1 January 1983
- journal article
- conference paper
- Published by AIP Publishing in Physics of Fluids
- Vol. 26 (1) , 14-29
- https://doi.org/10.1063/1.863994
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.Keywords
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