Dispersion of long gravity-capillary surface waves and asymptotic equations for solitons

Abstract

It is shown that in the description of long surface gravity-capillary waves, accounting for air density is essential when the fluid depth is close to the critical value. At critical depth the dispersive term of the third order (KdV-dispersion) in the Taylor's series of frequency on wavenumber vanishes due to expostulate actions of gravity and capillary effects. Estimates show that in the critical case the dispersive term of the second order (BO-dispersion), rather than the fifth order, as was thought before, becomes determinative. The main dispersive term, in the corresponding evolution equation for weakly nonlinear perturbations, determines a structure of solitary waves. Solitary wave solutions numerically obtained for the combined BO-KdV equation, as well as their Fourier spectra, are compared with the known soliton solutions of the BO and KdV equations.