We study the behaviour of shallow water waves propagating over bathymetry that varies periodically in one direction and is constant in the other. Plane waves travelling along the constant direction are known to evolve into solitary waves, due to an effective dispersion. We apply multiple-scale perturbation theory to derive an effective constant-coefficient system of equations, showing that the transversely averaged wave approximately satisfies a Boussinesq-type equation, while the lateral variation in the wave is related to certain integral functions of the bathymetry. Thus the homogenized equations not only accurately describe these waves but also predict their full two-dimensional shape in some detail. Numerical experiments confirm the good agreement between the effective equations and the variable-bathymetry shallow water equations.

This overview discusses the inverse scattering theory for the Kadomtsev–Petviashvili II equation, focusing on the inverse problem for perturbed multi-line solitons. Despite the introduction of new techniques to handle singularities, the theory remains consistent across various backgrounds, including the vacuum, 1-line and multi-line solitons.

The interaction of a solitary wave and a slowly varying mean background or flow for the Serre-Green-Naghdi (SGN) equations is studied using Whitham modulation theory. The exact form of the three SGN-Whitham modulation equations – two for the mean horizontal velocity and depth decoupled from one for the solitary wave amplitude field – is obtained in the solitary wave limit. Although the three equations are not diagonalizable, the restriction of the full system to simple waves for the mean equations is diagonalized in terms of Riemann invariants. The Riemann invariants are used to analytically describe the head-on and overtaking interactions of a solitary wave with a rarefaction wave and dispersive shock wave (DSW), leading to scenarios of solitary wave trapping or transmission by the mean flow. The analytical results for overtaking interactions prove that a simpler, approximate approach based on the DSW fitting method is accurate to the second order in solitary wave amplitude, beyond the first order accurate Korteweg-de Vries approximation. The analytical results also accurately predict the SGN DSW’s solitary wave edge amplitude and speed. The analytical results are favourably compared with corresponding numerical solutions of the full SGN equations. Because the SGN equations model the bi-directional propagation of strongly nonlinear, long gravity waves over a flat bottom, the analysis presented here describes large amplitudesolitary wave-mean flow interactions in shallow water waves.

The emitted internal wave groups and generating source at a double ridge-valley topography on the Norwegian Continental Shelf are determined. The location is 69 degrees north and 14 degrees east on the eastern boundary of the Norwegian Sea. Combination of two data sources – an ocean general circulation model and a set of satellite images – predicts the dominant shelf/slope current, the tide and the density stratification. The internal linear long-wave speed provides the reference velocity. The particular flow–topography interaction results in two compact internal tidal troughs, extending across the shelf, orthogonal to the current and separated by the diurnal internal tidal wavelength. The strongly nonlinear trough emits the wave groups advancing upstream at the diurnal frequency. Satellite data determine the spatial frequency and the number of groups. The dimensionless nonlinear excess propagation speed of 0.32 of the wave groups is compared to KdV theory and the model of internal solitary waves. Possible instability and supply of nutrients for a downstream cold-water coral reef are discussed. The data from satellite in combination with ocean model calculations at the mesoscale is general for the identification of nonlinear internal wave generation and propagation.

We consider the problem of computing a class of soliton gas primitive potentials for the Korteweg–de Vries equation that arise from the accumulation of solitons on an infinite interval in the physical domain, extending to $-\infty$. This accumulation results in an associated Riemann–Hilbert Problem (RHP) on a number of disjoint intervals. In the case where the jump matrices have specific square-root behaviour, we describe an efficient and accurate numerical method to solve this RHP and extract the potential. The keys to the method are, first, the deformation of the RHP, making numerical use of the so-called g-function, and, second, the incorporation of endpoint singularities into the chosen basis to discretize and solve the associated singular integral equation.