Global existence of small-norm solutions in the reduced Ostrovsky equation

We use a novel transformation of the reduced Ostrovsky equation to the integrable Tzitz\'eica equation and prove global existence of small-norm solutions in Sobolev space $H^3(R)$. This scenario is an alternative to finite-time wave breaking of large-norm solutions of the reduced Ostrovsky equation. We also discuss a sharp sufficient condition for the finite-time wave breaking.Comment: 11 pages; 1 figur

Fast and slow resonant triads in the two layer rotating shallow water equations

In this paper we examine triad resonances in a rotating shallow water system when there are two free interfaces. This allows for an examination in a relatively simple model of the interplay between baroclinic and barotropic dynamics in a context where there is also a geostrophic mode. In contrast to the much-studied one-layer rotating shallow water system, we find that as well as the usual slow geostrophic mode, there are now two fast waves, a barotropic mode and a baroclinic mode. This feature permits triad resonances to occur between three fast waves, with a mixture of barotropic and baroclinic modes, an aspect which cannot occur in the one-layer system. There are now also two branches of the slow geostrophic mode with a repeated branch of the dispersion relation. The consequences are explored in a derivation of the full set of triad interaction equations, using a multi-scale asymptotic expansion based on a small amplitude parameter. The derived nonlinear interaction coefficients are confirmed using energy and enstrophy conservation. These triad interaction equations are explored with an emphasis on the parameter regime with small Rossby and Froude numbers

"Dispersion management" for solitons in a Korteweg-de Vries system

The existence of ``dispersion-managed solitons'', i.e., stable pulsating solitary-wave solutions to the nonlinear Schr\"{o}dinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our purpose here is to investigate whether similar structures exist for other well-known nonlinear wave models. Hence, here we consider as a basic model the variable-coefficient Korteweg-de Vries equation; this has the form of a Korteweg-de Vries equation with a periodically varying third-order dispersion coefficient, that can take both positive and negative values. More generally, this model may be extended to include fifth-order dispersion. Such models may describe, for instance, periodically modulated waveguides for long gravity-capillary waves. We develop an analytical approximation for solitary waves in the weakly nonlinear case, from which it is possible to obtain a reduction to a relatively simple integral equation, which is readily solved numerically. Then, we describe some systematic direct simulations of the full equation, which use the soliton shape produced by the integral equation as an initial condition. These simulations reveal regions of stable and unstable pulsating solitary waves in the corresponding parametric space. Finally, we consider the effects of fifth-order dispersion.Comment: 19 pages, 7 figure

Internal solitary waves in a variable medium

In both the ocean and the atmosphere, the interaction of a density stratified flow with topography can generate large-amplitude, horizontally propagating internal solitary waves. Often these waves are observed in regions where the waveguide properties vary in the direction of propagation. In this article we consider nonlinear evolution equations of the Kortewegde Vries type, with variable coefficients, and use these models to review the properties of slowly-varying periodic and solitary waves

Nonlinear effects in wave scattering and generation

When a fluid flow interacts with a topographic feature, and the fluid can support wave propagation, then there is the potential for waves to be generated upstream and/or downstream. In many cases when the topographic feature has a small amplitude the situation can be successfully described using a linearised theory, and any nonlinear effects are determined as a small perturbation on the linear theory. However, when the flow is critical, that is, the system supports a long wave whose group velocity is zero in the reference frame of the topographic feature, then typically the linear theory fails and it is necessary to develop an intrinsically nonlinear theory. It is now known that in many cases such a transcritical, weakly nonlinear and weakly dispersive theory leads to a forced Korteweg- de Vries (fKdV) equation. In this article we shall sketch the contexts where the fKdV equation is applicable, and describe some of the most relevant solutions. There are two main classes of solutions. In the first, the initial condition for the fKdV equation is the zero state, so that the waves are generated directly by the flow interaction with the topography. In this case the solutions are characterised by the generation of upstream solitary waves and an oscillatory downstream wavetrain, with the detailed structure being determined by the detuning parameter and the polarity of the topographic forcing term. In the second class a solitary wave is incident on the topography, and depending on the system parameters may be repelled with a significant amplitude change, trapped with a change in amplitude, or allowed to pass by the topography with only a small change in amplitude

Models for instability in inviscid fluid flows, due to a resonance between two waves

In inviscid fluid flows instability arises generically due to a resonance between two wave modes. Here, it is shown that the structure of the weakly nonlinear regime depends crucially on whether the modal structure coincides, or remains distinct, at the resonance point where the wave phase speeds coincide. Then in the weakly nonlinear, long-wave limit the generic model consists either of a Boussinesq equation, or of two coupled Korteweg-de Vries equations, respectively. For short waves, the generic model is correspondingly either a nonlinear Klein-Gordon equation for the wave envelope, or a pair of coupled first-order envelope equations

Solitary wave solution for a non-integrable, variable coefficient nonlinear Schrodinger equation

A non-integrable, variable coefficient nonlinear Schrodinger equation which governs the nonlinear pulse propagation in an inhomogeneous medium is considered. The same equation is also applicable to optical pulse propagation in averaged, dispersion-managed optical fiber systems, or fiber systems with phase modulation and pulse compression. Multi-scale asymptotic techniques are employed to establish the leading order approximation of a solitary wave. A direct numerical simulation shows excellent agreement with the asymptotic solution. The interactions of two pulses are also studied

Solitary waves propagating over variable topography

Solitary water waves are long nonlinear waves that can propagate steadily over long distances. They were first observed by Russell in 1837 in a now famous report [26] on his observations of a solitary wave propagating along a Scottish canal, and on his subsequent experiments. Some forty years later theoretical work by Boussinesq [8] and Rayleigh [25] established an analytical model. Then in 1895 Korteweg and de Vries [21] derived the well-known equation which now bears their names. Significant further developments had to wait until the second half of the twentieth century, when there were two parallel developments. On the one hand it became realised that the Korteweg-de Vries equation was a valid model for solitary waves in a wide variety of physical contexts. On the other hand came the discovery of the soliton by Kruskal and Zabusky [27], with the subsequent rapid development of the modern theory of solitons and integrable systems

Hamiltonian-versus-energy diagrams in soliton theory

Parametric curves featuring Hamiltonian versus energy are useful in the theory of solitons in conservative nonintegrable systems with local nonlinearities. These curves can be constructed in various ways. We show here that it is possible to find the Hamiltonian (H) and energy (Q) for solitons of non-Kerr-law media with local nonlinearities without specific knowledge of the functional form of the soliton itself. More importantly, we show that the stability criterion for solitons can be formulated in terms of H and Q only. This allows us to derive all the essential properties of solitons based only on the concavity of the curve H vs Q. We give examples of these curves for various nonlinearity laws and show that they confirm the general principle. We also show that solitons of an unstable branch can transform into solitons of a stable branch by emitting small amplitude waves. As a result, we show that simple dynamics like the transformation of a soliton of an unstable branch into a soliton of a stable branch can also be predicted from the H-Q diagram