It is generally accepted that ocean internal solitary waves can arise from the interaction of the barotropic tide with the continental shelf, which generates an internal tide that in turn steepens and forms solitary waves as it propagates shorewards. Some field observations, however, reveal large-amplitude internal solitary waves in deep water, hundreds of kilometers away from the continental shelf, suggesting an alternative generation mechanism: tidal flow over steep topography forces a propagating beam of internal tidal wave energy which impacts the thermocline at a considerable distance from the forcing site and gives rise to internal solitary waves there. Motivated by this possibility, a simple nonlinear long-wave model is proposed for the interaction of a tidal wave beam with the thermocline and the ensuing local generation of solitary waves. The thermocline is modelled as a density jump across the interface of a shallow homogeneous fluid layer on top of a deep uniformly stratified fluid, and a finite-amplitude propagating internal wave beam of tidal frequency in the lower fluid is assumed to be incident and reflected at the interface. The induced weakly nonlinear long-wave disturbance on the interface is governed in the far field by an integral-differential equation which accounts for nonlinear and dispersive effects as well as energy loss owing to radiation into the lower fluid. Depending on the intensity of the incident beam, nonlinear wave steepening can overcome radiation damping so a series of solitary waves may arise in the thermocline. Sample numerical solutions of the governing evolution equation suggest that this mechanism is quite robust for typical oceanic conditions.

We consider the evolution of an initial disturbance described by the modified Korteweg-de Vries equation with a positive coefficient of the cubic nonlinear term, so that it can support solitons. Our primary aim is to determine the circumstances which can lead to the formation of solitons and/or breathers. We use the associated scattering problem and determine the discrete spectrum, where real eigenvalues describe solitons and complex eigenvalues describe breathers. For analytical convenience we consider various piecewise-constant initial conditions. We show how complex eigenvalues may be generated by bifurcation from either the real axis, or the imaginary axis; in the former case the bifurcation occurs as the unfolding of a double real eigenvalue. A bifurcation from the real axis describes the transition of a soliton pair with opposite polarities into a breather, while the bifurcation from the imaginary axis describes the generation of a breather from the continuous spectrum. Within the class of initial conditions we consider, a disturbance of one polarity, either positive or negative, will only generate solitons, and the number of solitons depends on the total mass. On the other hand, an initial disturbance with both polarities and very small mass will favor the generation of breathers, and the number of breathers then depends on the total energy. Direct numerical simulations of the modified Korteweg-de Vries equation confirms the analytical results, and show in detail the formation of solitons, breathers, and quasistationary coupled soliton pairs. Being based on spectral theory, our analytical results apply to the entire hierarchy of evolution equations connected with the same eigenvalue problem. (c) 2000 American Institute of Physics.

We consider the propagation of stratified fluid through a contraction near resonance, for which the governing asymptotic equation is the forced Korteweg-de Vries equation. Steady solutions of this equation are sought that have constant but differing amplitudes upstream and downstream of the contraction, and for which leading-order dispersive effects are retained. A numerical algorithm is outlined to obtain such solutions, yielding a parametric relationship between the normalized velocity perturbation and the normalized width perturbation. Matrix methods are then used to investigate the linear stability of these solutions.

The near-resonant flow of a stratified fluid through a localized contraction is considered in the long-wavelength weakly nonlinear limit to investigate the transient development of nonlinear internal waves and whether these might lead to local steady hydraulic flows. It is shown that under these circumstances the response of the fluid will fall into one of three categories, the first governed by a forced Korteweg–de Vries equation and the latter two by a variable-coefficient form of this equation. The variable-coefficient equation is discussed using analytical approximations and numerical solutions when the forcing is of the same (positive) and of opposite (negative) polarity to that of free solitary waves in the fluid. For positive and negative forcing, strong and weak resonant regimes will occur near the critical point. In these resonant regimes for positive forcing the flow becomes locally steady within the contraction, while for negative forcing it remains unsteady within the contraction. The boundaries of these resonant regimes are identified in the limits of long and short contractions, and for a number of common stratifications.

We present the results of a large number of careful numerical experiments carried out to investigate the way that the solution of the integrable focusing nonlinear Schr odinger equation with xed initial data, when taken to be close to the semiclassical limit, depends on analyticity properties of the data. In particular, we study a family of initial data that have complex singularities at an adjustable distance from the real axis. We also make use of a simple relation that provides the exact solution, at the centre of the wave eld, of the elliptic quasilinear system that appears formally as a leading-order model for the semi-classical dynamics. Among other things, we conclude that the semi-classical limit cannot be expected to be continuous with respect to the initial data, even for real analytic data, if there are certain complex singularities present. We argue that, in order to have well-posedness of the semiclassical limit, the correct setting is a physically relevant space of functions with compactly supported Fourier transforms.