The flow of a stratified fluid over topography is considered in the long-wavelength weakly nonlinear limit for the case when the flow is near resonance; that is, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. It is shown that the amplitude of this mode is governed by a forced Korteweg-de Vries equation. This equation is discussed both analytically and numerically for a variety of different cases, covering subcritical and supercritical flow, resonant or non-resonant, and for localized forcing that has either the same, or opposite, polarity to the solitary waves that would exist in the absence of forcing. In many cases a significant upstream disturbance is generated which consists of a train of solitary waves. The usefulness of internal hydraulic theory in interpreting the results is also demonstrated.

We consider unsteady undular bores for a pair of coupled equations of Boussinesqtype which contain the familiar fully nonlinear dissipationless shallow-water dynamics and the leading-order fully nonlinear dispersive terms. This system contains one horizontal space dimension and time and can be systematically derived from the full Euler equations for irrotational flows with a free surface using a standard long-wave asymptotic expansion. In this context the system was first derived by Su and Gardner. It coincides with the one-dimensional flat-bottom reduction of the Green-Naghdi 1 system and, additionally, has recently found a number of fluid dynamics applications other than the present context of shallow-water gravity waves. We then use the Whitham modulation theory for a one-phase periodic travelling wave to obtain an asymptotic analytical description of an undular bore in the Su-Gardner system for a full range of "depth" ratios across the bore. The positions of the leading and trailing edges of the undular bore and the amplitude of the leading solitary wave of the bore are found as functions of this "depth ratio". The formation of a partial undular bore with a rapidly-varying finite-amplitude trailing wave front is predicted for "depth ratios" across the bore exceeding 1.43. The analytical results from the modulation theory are shown to be in excellent agreement with full numerical solutions for the development of an undular bore in the Su-Gardner system. I INTRODUCTIONIn shallow water, the transition between two different basic states, each characterized by a constant depth and horizontal velocity, is usually referred to as a bore. For sufficiently large transitions, the front of the bore is often turbulent, but as noted in the classical work of Benjamin and Lighthill [1], transitions of moderate amplitude are accompanied by wave trains without any wave breaking, and are hence called undular bores. Well-known examples are the bores on the River Severn in England and the River Dordogne in France.Undular bores also arise in other fluid flow contexts; for instance they can occur as internal undular bores in the density-stratified waters of the coastal ocean (see, for instance, [2], [3]), and as striking wave-forms with associated cloud formation in the atmospheric boundary layer (see, for instance, [4], [5]). They can also arise in many other physical contexts, and in plasma physics for instance, are usually called collisionless shocks.The classical theory of shallow-water undular bores was initiated by Benjamin and Lighthill in [1]. It is based on the analysis of stationary solutions of the Kortewegde Vries (KdV) equation modified by a small viscous term [6]. Subsequent approaches to the same problem have been usually based on the Whitham modulation theory (see 2 [7], [8]), appropriately modified by dissipation; this allows one to study analytically the development of an undular bore to a steady state (see [9], [10], [11]). Most recently, this approach was used in [12] to study the development of an u...

The propagation of solitary waves, so-called nematicons, in a nonlinear nematic liquid crystal is considered in the nonlocal regime. Approximate modulation equations governing the evolution of input beams into steady nematicons are derived by using suitable trial functions in a Lagrangian formulation of the equations for a nematic liquid crystal. The variational equations are then extended to include the effect of diffractive loss as the beam evolves. It is found that the nonlocal nature of the interaction between the light and the nematic has a significant effect on the form of this diffractive radiation. Furthermore, it is this shed radiation that allows the input beam to evolve to a steady nematicon. Finally, excellent agreement is found between solutions of the modulation equations and numerical solutions of the nematic liquid-crystal equations. Disciplines Physical Sciences and Mathematics Publication DetailsMinzoni, A., Smyth, N. The propagation of solitary waves, so-called nematicons, in a nonlinear nematic liquid crystal is considered in the nonlocal regime. Approximate modulation equations governing the evolution of input beams into steady nematicons are derived by using suitable trial functions in a Lagrangian formulation of the equations for a nematic liquid crystal. The variational equations are then extended to include the effect of diffractive loss as the beam evolves. It is found that the nonlocal nature of the interaction between the light and the nematic has a significant effect on the form of this diffractive radiation. Furthermore, it is this shed radiation that allows the input beam to evolve to a steady nematicon. Finally, excellent agreement is found between solutions of the modulation equations and numerical solutions of the nematic liquid-crystal equations.

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The extended Korteweg-de Vries equation which includes nonlinear and dispersive terms cubic in the wave amplitude is derived from the water-wave equations and the Lagrangian for the water-wave equations. For the special case in which only the higher-order nonlinear term is retained, the extended Korteweg-de Vries equation is transformed into the Korteweg-de Vries equation. Modulation equations for this equation are then derived from the modulation equations for the Korteweg-de Vries equation and the undular bore solution of the extended Korteweg-de Vries equation is found as a simple wave solution of these modulation equations. The modulation equations are also used to extend the solution for the resonant flow of a fluid over topography. This resonant flow occurs when, in the weakly nonlinear, long-wave limit, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. In addition to the effect of higher-order terms, the effect of boundary-layer viscosity is also considered. These solutions (with and without viscosity) are compared with recent experimental and numerical results.