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.

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.

The travelling wave with the peaked profile is usually considered as a limit in the family of travelling waves with the smooth profiles. We study the linear and nonlinear stability of the peaked travelling wave by using a local model for shallow water waves, which is an extended version of the Hunter–Saxton equation. The evolution problem is well-defined in the function space $H^1_{\rm per} \cap W^{1,\infty}$, where we derive the linearised equations of motion and study the nonlinear evolution of co-periodic perturbations to the peaked periodic wave by using the method of characteristics. Within the linearised equations, we prove the spectral instability of the peaked travelling wave from the spectrum of the linearised operator in a Hilbert space, which completely covers the closed vertical strip with a specific half-width. Within the nonlinear equations, we prove the nonlinear instability of the peaked travelling wave by showing that the gradient of perturbations grows at the wave peak. By using numerical approximations of the smooth travelling waves and the spectrum of their associated linearised operator, we show that the spectral instability of the peaked travelling wave cannot be obtained as a limit in the family of the spectrally stable smooth travelling waves.

For Maxwell’s equations with nonlinear polarization we prove the existence of time-periodic breather solutions travelling along slab or cylindrical waveguides. The solutions are TE-modes which are localized in one (slab case) or both (cylindrical case) space directions orthogonal to the direction of propagation. We assume a magnetically inactive and electrically nonlinear material law with a linear $\chi^{(1)}$- and a cubic $\chi^{(3)}$-contribution to the polarization. The $\chi^{(1)}$-contribution may be retarded in time or instantaneous whereas the $\chi^{(3)}$-contribution is always assumed to be retarded in time. We consider two different cubic nonlinearities which provide a variational structure under suitable assumptions on the retardation kernels, in particular we require that for time-periodic solutions Maxwell’s equations are invariant under time-inversion. By choosing a sufficiently small propagation speed along the waveguide the second order formulation of the Maxwell system becomes essentially elliptic for the E-field so that solutions can be constructed by the mountain pass theorem. The compactness issues arising in the variational method are overcome by either the cylindrical geometry itself or by extra assumptions on the linear and nonlinear parts of the polarization in case of the slab geometry. Our approach to breather solutions in the presence of time-retardation is systematic in the sense that we look for general conditions on the Fourier-coefficients in time of the retardation kernels. Our main existence result is complemented by concrete examples of coefficient functions and retardation kernels.

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.

The nonlinear Schrödinger equation is a second-order nonlinear, integrable partial differential equation describing the propagation of nonlinear waves in a variety of media, including light propagation in optical fibres. Inspired by recently reported experiments, here we consider its generalization to higher, even orders, of derivatives corresponding in optics to higher orders of dispersion. We show that none of these equations are integrable and investigate the nature of singularities that cause the equations to fail the Painlevé test.

In this work, we carry out a rigorous analysis of a multi-soliton solution of the focusing nonlinear Schrödinger equation as the number, N, of solitons grows to infinity. We discover configurations of N-soliton solutions which exhibit the formation (as $N \to \infty$) of a soliton gas condensate. Specifically, we show that when the eigenvalues of the Zakharov–Shabat operator for the nonlinear Schrödinger equation accumulate on two bounded horizontal segments in the complex plane with norming constants bounded away from 0, then, asymptotically, the solution is described by a rapidly oscillatory elliptic-wave with constant velocity, on compact subsets of (x, t). We then consider more complex solutions with an extra soliton component, and we show that, in this deterministic setting, the kinetic theory of solitons applies. This is to be distinguished from previous analyses of soliton gases where the norming constants were tending to zero with N, and the asymptotic description only included elliptic waves in the long-time asymptotics.

A modification of the semi-empirical theory of stratified turbulent flow, which includes an equation for the density fluctuations (the potential energy of turbulence), is applied to describe the effect of internal gravity waves (IWs) on the small-scale turbulence. After considering the periodic IWs, special attention is paid to the action of internal solitons, such as the classical Gardner solitons and a strongly nonlinear solitary wave regularly observed in the Oregon Bay of the USA. It is confirmed that the presence of potential energy allows the existence of finite turbulence at any Richardson number.

We characterize a general traveling periodic wave of the defocusing mKdV (modified Korteweg–de Vries) equation by using a quotient of products of Jacobi’s elliptic theta functions. Compared to the standing periodic wave of the defocusing NLS (nonlinear Schrödinger) equation, these solutions are special cases of Riemann’s theta function of genus two. Based on our characterization, we derive a new two-parameter solution form which defines a general three-parameter solution form with the scaling transformation. Eigenfunctions of the Lax system for the general traveling periodic wave are also characterized as quotients of products of Jacobi’s theta functions. As the main outcome of our analytical computations, we derive a new solution of the defocusing mKdV equation which describes the kink breather propagating on a general traveling wave background.

We study discretized Landau–de Gennes gradient dynamics of finite lattices and graphs in the small intersite coupling regime (“anticontinuous limit”). We consider the case of $3 \times 3$ Q-tensor systems and extend recent results on small coupling intersite equilibria to the case of geometries without boundaries. We show that the equation for Landau–de Gennes equilibria is reduced to an $SO(3)-$equivariant equation on submanifolds that are diffeomorphic to products of projective planes and are parametrized by uniaxial Q-tensors. The gradient flow of the Landau–de Gennes energy has a normally hyperbolic invariant attracting submanifold that is also parametrized by uniaxial Q-tensors. We also present numerical studies of the Landau–de Gennes gradient flow in open and periodic chain geometries. We see a rapid approach to a near-uniaxial state at each site, as expected by the theory, and a much slower decay to an equilibrium configuration. The long time scale is several orders of magnitudes slower, and can depend on the size of the lattice and the initial condition. In the case of the circle we see evidence for two stable equilibria that are discrete analogues of curves belonging to the two homotopy classes of the projective plane. Evidence of bistability is also seen numerically in the open chain geometry.

We present an algorithm to compute the eigenvalues of the Broer–Kaup (BK) system. The BK system is a system of non-linear partial differential equations that can be used as a model for weakly non-linear wave phenomena in 1+1 dimensions, such as shallow water waves in a flume. It can be seen as the natural generalization of the more familiar Korteweg–de Vries (KdV) equation. Whereas the KdV equation is only valid for waves propagating in one direction, the BK system covers waves moving simultaneously in both directions. This makes the BK system the natural candidate model for reflection analysis in shallow water conditions. Analogous to the KdV equation, the eigenvalues of the BK system each characterize a soliton, which can be moving forward or backward. In this paper, we show how the eigenvalues can be computed numerically from free surface data. Under mild and verifiable conditions, we can apply quasi Sturm–Liouville oscillation theory to guarantee that every soliton is found.

This paper is devoted to a comprehensive analysis of a family of solutions of the focusing nonlinear Schrödinger equation called general rogue waves of infinite order. These solutions have recently been shown to describe various limit processes involving large-amplitude waves, and they have also appeared in some physical models not directly connected with nonlinear Schrödinger equations. We establish the following key property of these solutions: they are all in $L^2(\mathbb{R})$ with respect to the spatial variable but they exhibit anomalously slow temporal decay. In this paper, we define general rogue waves of infinite order, establish their basic exact and asymptotic properties, and provide computational tools for calculating them accurately.

We investigate the long-time asymptotic behavior of solutions to the Cauchy problem for the KdV equation, focusing on the evolution of the radiant wave associated with a Wigner–von Neumann (WvN) resonance induced by the initial data (potential). A WvN resonance refers to an energy level where the potential exhibits zero transmission (complete reflection). The corresponding Jost solution at such energy becomes singular, and in the NLS context, this is referred to as a spectral singularity. A WvN resonance represents a long-range phenomenon, often introducing significant challenges, such as an infinite negative spectrum, when employing the inverse scattering transform (IST). To avoid some of these issues, we consider a restricted class of initial data that generates a WvN resonance but for which the IST framework can be suitably adapted. For this class of potentials, we demonstrate that each WvN resonance produces a distinct asymptotic regime – termed the resonance regime – characterized by a slower decay rate for large time compared to the radiant waves associated with short-range initial data.

Substantially extending previous results of the authors for smooth solutions in the viscous case, we develop linear damping estimates for periodic roll-wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws. Such damping estimates, consisting of $H^s$ energy estimates yielding exponential slaving of high-derivative to low-derivative norms, have served as crucial ingredients in nonlinear stability analyses of traveling waves in hyperbolic or partially parabolic systems, both in obtaining high-frequency resolvent estimates and in closing a nonlinear iteration for which available linearized stability estimates apparently lose regularity. Here, we establish for systems of size $n\leq 6$ a Lyapunov-type theorem stating that such energy estimates are available whenever strict high-frequency spectral stability holds; for dimensions $7$ and higher, there may be in general a gap between high-frequency spectral stability and existence of the type of energy estimate that we develop here. A key ingredient is a dimension-dependent linear algebraic lemma reminiscent of Lyapunov’s Lemma for ODE that is to our knowledge new.