995 research outputs found
Transformation of a shoaling undular bore
We consider the propagation of a shallow-water undular bore over a gentle monotonic bottom slope connecting two regions of constant depth, in the framework of the variable-coefficient Korteweg-de Vries equation. We show that, when the undular bore advances in the direction of decreasing depth, its interaction with the slowly varying topography results, apart from an adiabatic deformation of the bore itself, in the generation of a sequence of isolated solitons - an expanding large-amplitude modulated solitary wavetrain propagating ahead of the bore. Using nonlinear modulation theory we construct an asymptotic solution describing the formation and evolution of this solitary wavetrain. Our analytical solution is supported by direct numerical simulations. The presented analysis can be extended to other systems describing the propagation of undular bores (dispersive shock waves) in weakly non-uniform environments
Numerical study on diverging probability density function of flat-top solitons in an extended Korteweg-de Vries equation
We consider an extended Korteweg-de Vries (eKdV) equation, the usual
Korteweg-de Vries equation with inclusion of an additional cubic nonlinearity.
We investigate the statistical behaviour of flat-top solitary waves described
by an eKdV equation in the presence of weak dissipative disorder in the linear
growth/damping term. With the weak disorder in the system, the amplitude of
solitary wave randomly fluctuates during evolution. We demonstrate numerically
that the probability density function of a solitary wave parameter
which characterizes the soliton amplitude exhibits loglognormal divergence near
the maximum possible value.Comment: 8 pages, 4 figure
Transcritical flow of a stratified fluid: The forced extended Korteweg-de Vries model
Transcritical, or resonant, flow of a stratified fluid over an obstacle is studied using a forced extended Korteweg-de Vries model. This model is particularly relevant for a two-layer fluid when the layer depths are near critical, but can also be useful in other similar circumstances. Both quadratic and cubic nonlinearities are present and they are balanced by third-order dispersion. We consider both possible signs for the cubic nonlinear term but emphasize the less-studied case when the cubic nonlinear term and the dispersion term have the same-signed coefficients. In this case, our numerical computations show that two kinds of solitary waves are found in certain parameter regimes. One kind is similar to those of the well-known forced Korteweg-de Vries model and occurs when the cubic nonlinear term is rather small, while the other kind is irregularly generated waves of variable amplitude, which may continually interact. To explain this phenomenon, we develop a hydraulic theory in which the dispersion term in the model is omitted. This theory can predict the occurence of upstream and downstream undular bores, and these predictions are found to agree quite well with the numerical computations. © 2002 American Institute of Physics.published_or_final_versio
Cuspons, peakons and regular gap solitons between three dispersion curves
A general wave model with the cubic nonlinearity is introduced to describe a
situation when the linear dispersion relation has three branches, which would
intersect in the absence of linear couplings between the three waves. Actually,
the system contains two waves with a strong linear coupling between them, to
which a third wave is then coupled. This model has two gaps in its linear
spectrum. Realizations of this model can be made in terms of temporal or
spatial evolution of optical fields in, respectively, a planar waveguide or a
bulk-layered medium resembling a photonic-crystal fiber. Another physical
system described by the same model is a set of three internal wave modes in a
density-stratified fluid. A nonlinear analysis is performed for solitons which
have zero velocity in the reference frame in which the group velocity of the
third wave vanishes. Disregarding the self-phase modulation (SPM) term in the
equation for the third wave, we find two coexisting families of solitons:
regular ones, which may be regarded as a smooth deformation of the usual gap
solitons in a two-wave system, and cuspons with a singularity in the first
derivative at their center. Even in the limit when the linear coupling of the
third wave to the first two vanishes, the soliton family remains drastically
different from that in the linearly uncoupled system; in this limit, regular
solitons whose amplitude exceeds a certain critical value are replaced by
peakons. While the regular solitons, cuspons, and peakons are found in an exact
analytical form, their stability is tested numerically, which shows that they
all may be stable. If the SPM terms are retained, we find that there again
coexist two different families of generic stable soliton solutions, namely,
regular ones and peakons.Comment: a latex file with the text and 10 pdf files with figures. Physical
Review E, in pres
Steady transcritical flow over an obstacle: Parametric map of solutions of the forced extended Korteweg-de Vries equation
Transcritical flow of a stratified fluid over an obstacle is often modeled by the forced Korteweg-de Vries equation, which describes a balance among weak nonlinearity, weak dispersion, and small forcing effects. However, in some special circumstances, it is necessary to add an additional cubic nonlinear term, so that the relevant model is the forced extended Korteweg-de Vries equation. Here we seek steady solutions with constant, but different amplitudes upstream and downstream of the forcing region. Our main interest is in the case when the forcing has negative polarity, which represents a hole. The effects of the width of the hole and the amplitude of the hole on these steady solutions are then investigated. © 2011 American Institute of Physics.published_or_final_versio
Transformation of Quartz to Tridymite in the Presence of Binary Silicate Liquids
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/66429/1/j.1151-2916.1967.tb15146.x.pd
Nonlinear interfacial waves in a constant-vorticity planar flow over variable depth
Exact Lagrangian in compact form is derived for planar internal waves in a
two-fluid system with a relatively small density jump (the Boussinesq limit
taking place in real oceanic conditions), in the presence of a background shear
current of constant vorticity, and over arbitrary bottom profile. Long-wave
asymptotic approximations of higher orders are derived from the exact
Hamiltonian functional in a remarkably simple way, for two different
parametrizations of the interface shape.Comment: revtex, 4.5 pages, minor corrections, summary added, accepted to JETP
Letter
Unsteady undular bores in fully nonlinear shallow-water theory
We consider unsteady undular bores for a pair of coupled equations of
Boussinesq-type 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 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.Comment: Revised version accepted for publication in Phys. Fluids, 51 pages, 9
figure
Brief communication: Modulation instability of internal waves in a smoothly stratified shallow fluid with a constant buoyancy frequency
Unexpectedly large displacements in the interior of the
oceans are studied through the dynamics of packets of internal waves, where
the evolution of these displacements is governed by the nonlinear Schrödinger equation.
In cases with a constant buoyancy frequency, analytical treatment can be performed. While
modulation instability in surface wave packets only arises for sufficiently
deep water, “rogue” internal waves may occur in shallow water and
intermediate depth regimes. A dependence on the stratification parameter
and the choice of internal modes can be demonstrated explicitly. The spontaneous
generation of rogue waves is tested here via numerical simulation.</p
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