Generation of undular bores in the shelves of slowly-varying solitary waves

We study the long-time evolution of the trailing shelves that form behind solitary waves moving through an inhomogeneous medium, within the framework of the variable-coefficient Korteweg-de Vries equation. We show that the nonlinear evolution of the shelf leads typically to the generation of an undular bore and an expansion fan, which form apart but start to overlap and nonlinearly interact after a certain time interval. The interaction zone expands with time and asymptotically as time goes to infinity occupies the whole perturbed region. Its oscillatory structure strongly depends on the sign of the inhomogeneity gradient of the variable background medium. We describe the nonlinear evolution of the shelves in terms of exact solutions to the KdV-Whitham equations with natural boundary conditions for the Riemann invariants. These analytic solutions, in particular, describe the generation of small "secondary" solitary waves in the trailing shelves, a process observed earlier in various numerical simulations

Coupled Ostrovsky equations for internal waves in a shear flow

In the context of fluid flows, the coupled Ostrovsky equations arise when two distinct linear long wave modes have nearly coincident phase speeds in the presence of background rotation. In this paper, nonlinear waves in a stratified fluid in the presence of shear flow are investigated both analytically, using techniques from asymptotic perturbation theory, and through numerical simulations. The dispersion relation of the system, based on a three-layer model of a stratified shear flow, reveals various dynamical behaviours, including the existence of unsteady and steady envelope wave packets.Comment: 47 pages, 39 figures, accepted to Physics of Fluid

Transcritical shallow-water flow past topography: finite-amplitude theory

We consider shallow-water flow past a broad bottom ridge, localized in the flow direction, using the framework of the forced SuGardner (SG) system of equations, with a primary focus on the transcritical regime when the Froude number of the oncoming flow is close to unity. These equations are an asymptotic long-wave approximation of the full Euler system, obtained without a simultaneous expansion in the wave amplitude, and hence are expected to be superior to the usual weakly nonlinear Boussinesq-type models in reproducing the quantitative features of fully nonlinear shallow-water flows. A combination of the local transcritical hydraulic solution over the localized topography, which produces upstream and downstream hydraulic jumps, and unsteady undular bore solutions describing the resolution of these hydraulic jumps, is used to describe various flow regimes depending on the combination of the topography height and the Froude number. We take advantage of the recently developed modulation theory of SG undular bores to derive the main parameters of transcritical fully nonlinear shallow-water flow, such as the leading solitary wave amplitudes for the upstream and downstream undular bores, the speeds of the undular bores edges and the drag force. Our results confirm that most of the features of the previously developed description in the framework of the unidirectional forced Kortewegde Vries (KdV) model hold up qualitatively for finite amplitude waves, while the quantitative description can be obtained in the framework of the bidirectional forced SG system. Our analytic solutions agree with numerical simulations of the forced SG equations within the range of applicability of these equations

Transformation of a shoaling undular bore

We consider the propagation of a shallow-water undular bore over a gentle monotonic bottom slope connecting two regions of constant depth, in the framework of the variable-coefficient Korteweg-de Vries equation. We show that, when the undular bore advances in the direction of decreasing depth, its interaction with the slowly varying topography results, apart from an adiabatic deformation of the bore itself, in the generation of a sequence of isolated solitons - an expanding large-amplitude modulated solitary wavetrain propagating ahead of the bore. Using nonlinear modulation theory we construct an asymptotic solution describing the formation and evolution of this solitary wavetrain. Our analytical solution is supported by direct numerical simulations. The presented analysis can be extended to other systems describing the propagation of undular bores (dispersive shock waves) in weakly non-uniform environments

Wave Breaking and the Generation of Undular Bores in an Integrable Shallow Water System

The generation of an undular bore in the vicinity of a wave‐breaking point is considered for the integrable Kaup–Boussinesq (KB) shallow water system. In the framework of the Whitham modulation theory, an analytic solution of the Gurevich–Pitaevskii type of problem for a generic “cubic” breaking regime is obtained using a generalized hodograph transform, and a further reduction to a linear Euler–Poisson equation. The motion of the undular bore edges is investigated in detail

Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction

This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variable-coefficient Korteweg-de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adiabatic evolution of isolated solitary waves and periodic wave trains in the presence of variable topography and bottom friction, modelled by the Chezy law, but also, importantly, to study the effects of these factors on the propagation of undular bores, which are essentially unsteady in the system under consideration. In particular, it is shown that the combined action of variable topography and bottom friction generally imposes certain global restrictions on the undular bore propagation so that the evolution of the leading solitary wave can be substantially different from that of an isolated solitary wave with the same initial amplitude. This non-local effect is due to nonlinear wave interactions within the undular bore and can lead to an additional solitary wave amplitude growth, which cannot be predicted in the framework of the traditional adiabatic approach to the propagation of solitary waves in slowly varying media

Soliton formation from a pulse passing the zero-dispersion point in a nonlinear Schr\"odinger equation

We consider in detail the self-trapping of a soliton from a wave pulse that passes from a defocussing region into a focussing one in a spatially inhomogeneous nonlinear waveguide, described by a nonlinear Schrodinger equation in which the dispersion coefficient changes its sign from normal to anomalous. The model has direct applications to dispersion-decreasing nonlinear optical fibers, and to natural waveguides for internal waves in the ocean. It is found that, depending on the (conserved) energy and (nonconserved) mass of the initial pulse, four qualitatively different outcomes of the pulse transformation are possible: decay into radiation; self-trapping into a single soliton; formation of a breather; and formation of a pair of counterpropagating solitons. A corresponding chart is drawn on a parametric plane, which demonstrates some unexpected features. In particular, it is found that any kind of soliton(s) (including the breather and counterpropagating pair) eventually decays into pure radiation with the increase of the energy, the initial mass being kept constant. It is also noteworthy that a virtually direct transition from a single soliton into a pair of symmetric counterpropagating ones seems possible. An explanation for these features is proposed. In two cases when analytical approximations apply, viz., a simple perturbation theory for broad initial pulses, or the variational approximation for narrow ones, comparison with the direct simulations shows reasonable agreement.Comment: 18 pages, 10 figures, 1 table. Phys. Rev. E, in pres

The Modulation of Multiple Phases Leading to the Modified KdV Equation

This paper seeks to derive the modified KdV (mKdV) equation using a novel approach from systems generated from abstract Lagrangians that possess a two-parameter symmetry group. The method to do uses a modified modulation approach, which results in the mKdV emerging with coefficients related to the conservation laws possessed by the original Lagrangian system. Alongside this, an adaptation of the method of Kuramoto is developed, providing a simpler mechanism to determine the coefficients of the nonlinear term. The theory is illustrated using two examples of physical interest, one in stratified hydrodynamics and another using a coupled Nonlinear Schr\"odinger model, to illustrate how the criterion for the mKdV equation to emerge may be assessed and its coefficients generated.Comment: 35 pages, 5 figure

Cuspons, peakons and regular gap solitons between three dispersion curves

A general wave model with the cubic nonlinearity is introduced to describe a situation when the linear dispersion relation has three branches, which would intersect in the absence of linear couplings between the three waves. Actually, the system contains two waves with a strong linear coupling between them, to which a third wave is then coupled. This model has two gaps in its linear spectrum. Realizations of this model can be made in terms of temporal or spatial evolution of optical fields in, respectively, a planar waveguide or a bulk-layered medium resembling a photonic-crystal fiber. Another physical system described by the same model is a set of three internal wave modes in a density-stratified fluid. A nonlinear analysis is performed for solitons which have zero velocity in the reference frame in which the group velocity of the third wave vanishes. Disregarding the self-phase modulation (SPM) term in the equation for the third wave, we find two coexisting families of solitons: regular ones, which may be regarded as a smooth deformation of the usual gap solitons in a two-wave system, and cuspons with a singularity in the first derivative at their center. Even in the limit when the linear coupling of the third wave to the first two vanishes, the soliton family remains drastically different from that in the linearly uncoupled system; in this limit, regular solitons whose amplitude exceeds a certain critical value are replaced by peakons. While the regular solitons, cuspons, and peakons are found in an exact analytical form, their stability is tested numerically, which shows that they all may be stable. If the SPM terms are retained, we find that there again coexist two different families of generic stable soliton solutions, namely, regular ones and peakons.Comment: a latex file with the text and 10 pdf files with figures. Physical Review E, in pres