311 research outputs found
Shear induced breaking of large internal solitary waves
The stability properties of 24 experimentally generated internal solitary waves (ISWs) of extremely large amplitude, all with minimum Richardson number less than 1/4, are investigated. The study is supplemented by fully nonlinear calculations in a three-layer fluid. The waves move along a linearly stratified pycnocline (depth h2) sandwiched between a thin upper layer (depth h1) and a deep lower layer (depth h3), both homogeneous. In particular, the wave-induced velocity profile through the pycnocline is measured by particle image velocimetry (PIV) and obtained in computation. Breaking ISWs were found to have amplitudes (a1) in the range a1>2.24 āh1h2(1+h2/h1), while stable waves were on or below this limit. Breaking ISWs were investigated for 0.27 0.86 and stable waves for Lx/Ī» < 0.86. The results show a sort of threshold-like behaviour in terms of Lx/Ī». The results demonstrate that the breaking threshold of Lx/Ī» = 0.86 was sharper than one based on a minimum Richardson number and reveal that the Richardson number was found to become almost antisymmetric across relatively thick pycnoclines, with the minimum occurring towards the top part of the pycnoclinePostprintPeer reviewe
Generalised Fourier Transform and Perturbations to Soliton Equations
A brief survey of the theory of soliton perturbations is presented. The focus
is on the usefulness of the so-called Generalised Fourier Transform (GFT). This
is a method that involves expansions over the complete basis of `squared
olutions` of the spectral problem, associated to the soliton equation. The
Inverse Scattering Transform for the corresponding hierarchy of soliton
equations can be viewed as a GFT where the expansions of the solutions have
generalised Fourier coefficients given by the scattering data.
The GFT provides a natural setting for the analysis of small perturbations to
an integrable equation: starting from a purely soliton solution one can
`modify` the soliton parameters such as to incorporate the changes caused by
the perturbation.
As illustrative examples the perturbed equations of the KdV hierarchy, in
particular the Ostrovsky equation, followed by the perturbation theory for the
Camassa- Holm hierarchy are presented.Comment: 20 pages, no figures, to appear in: Discrete and Continuous Dynamical
Systems
Oleinik type estimates for the Ostrovsky-Hunter eequation
The Ostrovsky-Hunter equation provides a model for small-amplitude long waves
in a rotating fluid of finite depth. It is a nonlinear evolution equation. In
this paper we study the well-posedness for the Cauchy problem associated to
this equation within a class of bounded discontinuous solutions. We show that
we can replace the Kruzkov-type entropy inequalities by an Oleinik-type
estimate and prove uniqueness via a nonlocal adjoint problem. An implication is
that a shock wave in an entropy weak solution to the Ostrovsky-Hunter equation
is admissible only if it jumps down in value (like the inviscid Burgers
equation)
Orbital stability of periodic waves in the class of reduced Ostrovsky equations
Periodic travelling waves are considered in the class of reduced Ostrovsky
equations that describe low-frequency internal waves in the presence of
rotation. The reduced Ostrovsky equations with either quadratic or cubic
nonlinearities can be transformed to integrable equations of the Klein--Gordon
type by means of a change of coordinates. By using the conserved momentum and
energy as well as an additional conserved quantity due to integrability, we
prove that small-amplitude periodic waves are orbitally stable with respect to
subharmonic perturbations, with period equal to an integer multiple of the
period of the wave. The proof is based on construction of a Lyapunov
functional, which is convex at the periodic wave and is conserved in the time
evolution. We also show numerically that convexity of the Lyapunov functional
holds for periodic waves of arbitrary amplitudes.Comment: 34 page
Existence, regularity, and symmetry of periodic traveling waves for Gardner-Ostrovsky type equations
We study the existence, regularity, and symmetry of periodic traveling
solutions to a class of Gardner-Ostrovsky type equations, including the
classical Gardner-Ostrovsky equation, the (modified) Ostrovsky, and the reduced
(modified) Ostrovsky equation. The modified Ostrovsky equation is also known as
the short pulse equation. The Gardner-Ostrovsky equation is a model for
internal ocean waves of large amplitude. We prove the existence of nontrivial,
periodic traveling wave solutions using local bifurcation theory, where the
wave speed serves as the bifurcation parameter. Moreover, we give a regularity
analysis for periodic traveling solutions in the presence as well as absence of
Boussinesq dispersion. We see that the presence of Boussinesq dispersion
implies smoothness of periodic traveling wave solutions, while its absence may
lead to singularities in the form of peaks or cusps. Eventually, we study the
symmetry of periodic traveling solutions by the method of moving planes. A
novel feature of the symmetry results in the absence of Boussinesq dispersion
is that we do not need to impose a traditional monotonicity condition or a
recently developed reflection criterion on the wave profiles to prove the
statement on the symmetry of periodic traveling waves
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