232 research outputs found
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 . 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
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
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
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
Generation of Secondary Solitary Waves in the Variable-Coefficient Korteweg-de Vries Equation
We consider the solitary wave solutions of a Korteweg-de Vries equation, where the
coefficients in the equation vary with time over a certain region. When these coefficients
vary rapidly compared with the solitary wave, then it is well-known that the solitary wave
may fission into two or more solitary waves. On the other hand, when these coefficients vary
slowly, the solitary wave deforms adiabatically with the production of a trailing shelf. In
this paper we re-examine this latter case, and show that the trailing shelf, on a very long
time-scale, can lead to the generation of small secondary solitary waves. This result thus
provides a connection between the adiabatic deformation regime, and the fission regime
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