On the Surface Waves in a Shallow Channel with an Uneven Bottom

Abstract

This paper examines surface gravity waves in a shallow channel with periodic or random bottom irregularities. Three types of bottom topography are distinguished, allowing a simplification of the basic shallow-water-wave equations. For two of them, asymptotic equations of the Korteveg-de Vries type are derived (the third type has been considered earlier by other authors).

Stability and weak rotation limit of solitary waves of the Ostrovsky equation

“…The parameter β determines the type of dispersion. For β < 0 (negative dispersion), the equation models surface and internal waves in the ocean and surface waves in a shallow channel with uneven bottom [2], while for β > 0 (positive dispersion), it models capillary waves on the surface of a liquid and magneto-acoustic waves in a plasma [4,5]. Liu and Varlamov [14] Liu and Varlamov [9] showed using the Concentration Compactness Lemma that there exist solutions of (1.2) in the space X 1 provided c < 2 √ βγ.…”

Stability of Solitary Waves of a Generalized Ostrovsky Equation

“…The parameter β determines the type of dispersion, namely, β < 0 (negative dispersion) for surface and internal waves in the ocean or surface waves in a shallow channel with an uneven bottom and β > 0 (positive dispersion) for capillary waves on the surface of liquid or for oblique magneto-acoustic waves in plasma. See Benilov [2], Galkin, Stepanyants and Gilman [7] and Gilman, Grimshaw and Stepanyants [8]. Considered herein is the generalization of the Ostrovsky equation…”

Low-Regularity Solutions for the Ostrovsky Equation

“…(ii) β = 1 (positive dispersion) in the cases of capillary waves on the surface of a liquid or oblique magneto-acoustic waves in plasma [1,5,6].…”