Phase and amplitude characteristics of nonlinear dispersion models improved precision

The properties of the dispersion relations for two new fully nonlinear weakly dispersive shallow water models are studied, for which, with certain parameters, it is possible to obtain the fourth, sixth, or eighth order of accuracy of the approximation of the phase velocity of the three-dimensional potential current model. For the hierarchy of shallow water models, under the assumption of a slightly changing shape of the bottom, formulas are obtained that establish the relationship between the rate of change in the wave amplitude and the rate of change in the thickness of the liquid layer, and the dependences of the amplitude and length of the incident wave on the depth of the water area are also derived. It is shown that the new model of the fourth order long-wavelength approximation with the eighth order of accuracy of the dispersion relation provides the best approximation of the considered characteristics in the case of both a horizontal bottom and a variable-shaped bottom.