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.

Linear and integrable non-linear fractional evolution equations are discussed. Earlier results for the integrable fractional Korteweg–deVries (KdV) equation and the KdV hierarchy are reviewed. Using these as a guide, the fractional integrable Burgers equation and hierarchy and its solutions are analysed. Some explicit solutions are provided.

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.

We prove the existence of small-amplitude periodic travelling waves in dimer Fermi–Pasta–Ulam–Tsingou (FPUT) lattices without assumptions of physical symmetry. Such lattices are infinite, one-dimensional chains of coupled particles in which the particle masses and/or the potentials of the coupling springs can alternate. Previously, periodic travelling waves were constructed in a variety of limiting regimes for the symmetric mass and spring dimers, in which only one kind of material data alternates. The new results discussed here remove the symmetry assumptions by exploiting the gradient structure and translation invariance of the travelling wave problem. Together, these features eliminate certain solvability conditions that symmetry would otherwise manage and facilitate a bifurcation argument involving a two-dimensional kernel and cokernel.

Burgers equation is a classic model, which arises in numerous applications. At its very core, it is a simple conservation law, which serves as a toy model for various dynamics phenomena. In particular, it supports explicit heteroclinic solutions, both fronts and backs. Their stability has been studied in detail. There has been substantial interest in considering dispersive and/or diffusive modifications, which present novel dynamical paradigms in such a simple setting. More specifically, the KdV–Burgers model has been shown to support unique fronts (not all of them monotone!) with fixed values at $\pm \infty$. Many articles, among which [11], [9] and [10], have studied the question of stability of monotone (or close to monotone) fronts. In a breakthrough paper [2], the authors have extended these results in several different directions. They have considered a wider range of models. The fronts do not need to be monotone but are subject of a spectral condition instead. Most importantly, the method allows for large perturbations, as long as the heteroclinic conditions at $\pm \infty$ are met. That is, there is asymptotic attraction to the said fronts or equivalently the limit set consists of one point. The purpose of this paper is to extend the results of [2] by providing explicit algebraic rates of convergence as $t\to \infty$. We bootstrap these results from the results in [2] using additional energy estimates for two important examples, namely KdV–Burgers and the fractional Burgers problem. These rates are likely not optimal, but we conjecture that they are algebraic nonetheless.