526 research outputs found
Collisions of acoustic solitons and their electric fields in plasmas at critical compositions
Acoustic solitons obtained through a reductive perturbation scheme are
normally governed by a Korteweg-de Vries (KdV) equation. In multispecies
plasmas at critical compositions the coefficient of the quadratic nonlinearity
vanishes. Extending the analytic treatment then leads to a modified KdV (mKdV)
equation, which is characterized by a cubic nonlinearity and is even in the
electrostatic potential. The mKdV equation admits solitons having opposite
electrostatic polarities, in contrast to KdV solitons which can only be of one
polarity at a time. A Hirota formalism has been used to derive the two-soliton
solution. That solution covers not only the interaction of same-polarity
solitons but also the collision of compressive and rarefactive solitons. For
the visualisation of the solutions, the focus is on the details of the
interaction region. A novel and detailed discussion is included of typical
electric field signatures that are often observed in ionospheric and
magnetospheric plasmas. It is argued that these signatures can be attributed to
solitons and their interactions. As such, they have received little attention.Comment: 15 pages, 15 figure
Solitons and kinks in a general car-following model
We study a car-following model of traffic flow which assumes only that a
car's acceleration depends on its own speed, the headway ahead of it, and the
rate of change of headway, with only minimal assumptions about the functional
form of that dependence. The velocity of uniform steady flow is found
implicitly from the acceleration function, and its linear stability criterion
can be expressed simply in terms of it. Crucially, unlike in previously
analyzed car-following models, the threshold of absolute stability does not
generally coincide with an inflection point in the steady velocity function.
The Burgers and KdV equations can be derived under the usual assumptions, but
the mKdV equation arises only when absolute stability does coincide with an
inflection point. Otherwise, the KdV equation applies near absolute stability,
while near the inflection point one obtains the mKdV equation plus an extra,
quadratic term. Corrections to the KdV equation "select" a single member of the
one-parameter set of soliton solutions. In previous models this has always
marked the threshold of a finite- amplitude instability of steady flow, but
here it can alternatively be a stable, small-amplitude jam. That is, there can
be a forward bifurcation from steady flow. The new, augmented mKdV equation
which holds near an inflection point admits a continuous family of kink
solutions, like the mKdV equation, and we derive the selection criterion
arising from the corrections to this equation.Comment: 25 page
Internal solitary waves in the ocean: Analysis using the periodic, inverse scattering transform
The periodic, inverse scattering transform (PIST) is a powerful analytical
tool in the theory of integrable, nonlinear evolution equations. Osborne
pioneered the use of the PIST in the analysis of data form inherently nonlinear
physical processes. In particular, Osborne's so-called nonlinear Fourier
analysis has been successfully used in the study of waves whose dynamics are
(to a good approximation) governed by the Korteweg--de Vries equation. In this
paper, the mathematical details and a new application of the PIST are
discussed. The numerical aspects of and difficulties in obtaining the nonlinear
Fourier (i.e., PIST) spectrum of a physical data set are also addressed. In
particular, an improved bracketing of the "spectral eigenvalues" (i.e., the
+/-1 crossings of the Floquet discriminant) and a new root-finding algorithm
for computing the latter are proposed. Finally, it is shown how the PIST can be
used to gain insightful information about the phenomenon of soliton-induced
acoustic resonances, by computing the nonlinear Fourier spectrum of a data set
from a simulation of internal solitary wave generation and propagation in the
Yellow Sea.Comment: 10 pages, 4 figures (6 images); v2: corrected a few minor mistakes
and typos, version accepted for publication in Math. Comput. Simu
Hidden solitons in the Zabusky-Kruskal experiment: Analysis using the periodic, inverse scattering transform
Recent numerical work on the Zabusky--Kruskal experiment has revealed,
amongst other things, the existence of hidden solitons in the wave profile.
Here, using Osborne's nonlinear Fourier analysis, which is based on the
periodic, inverse scattering transform, the hidden soliton hypothesis is
corroborated, and the \emph{exact} number of solitons, their amplitudes and
their reference level is computed. Other "less nonlinear" oscillation modes,
which are not solitons, are also found to have nontrivial energy contributions
over certain ranges of the dispersion parameter. In addition, the reference
level is found to be a non-monotone function of the dispersion parameter.
Finally, in the case of large dispersion, we show that the one-term nonlinear
Fourier series yields a very accurate approximate solution in terms of Jacobian
elliptic functions.Comment: 10 pages, 4 figures (9 images); v2: minor revision, version accepted
for publication in Math. Comput. Simula
Tau-Functions and Generalized Integrable Hierarchies
The tau-function formalism for a class of generalized ``zero-curvature''
integrable hierarchies of partial differential equations, is constructed. The
class includes the Drinfel'd-Sokolov hierarchies. A direct relation between the
variables of the zero-curvature formalism and the tau-functions is established.
The formalism also clarifies the connection between the zero-curvature
hierarchies and the Hirota-type hierarchies of Kac and Wakimoto.Comment: 23 page
Dark solitons in atomic Bose-Einstein condensates: from theory to experiments
This review paper presents an overview of the theoretical and experimental
progress on the study of matter-wave dark solitons in atomic Bose-Einstein
condensates. Upon introducing the general framework, we discuss the statics and
dynamics of single and multiple matter-wave dark solitons in the quasi
one-dimensional setting, in higher-dimensional settings, as well as in the
dimensionality crossover regime. Special attention is paid to the connection
between theoretical results, obtained by various analytical approaches, and
relevant experimental observations.Comment: 82 pages, 13 figures. To appear in J. Phys. A: Math. Theor
Nonlinear theory of solitary waves associated with longitudinal particle motion in lattices - Application to longitudinal grain oscillations in a dust crystal
The nonlinear aspects of longitudinal motion of interacting point masses in a
lattice are revisited, with emphasis on the paradigm of charged dust grains in
a dusty plasma (DP) crystal. Different types of localized excitations,
predicted by nonlinear wave theories, are reviewed and conditions for their
occurrence (and characteristics) in DP crystals are discussed. Making use of a
general formulation, allowing for an arbitrary (e.g. the Debye electrostatic or
else) analytic potential form and arbitrarily long site-to-site range
of interactions, it is shown that dust-crystals support nonlinear kink-shaped
localized excitations propagating at velocities above the characteristic DP
lattice sound speed . Both compressive and rarefactive kink-type
excitations are predicted, depending on the physical parameter values, which
represent pulse- (shock-)like coherent structures for the dust grain relative
displacement. Furthermore, the existence of breather-type localized
oscillations, envelope-modulated wavepackets and shocks is established. The
relation to previous results on atomic chains as well as to experimental
results on strongly-coupled dust layers in gas discharge plasmas is discussed.Comment: 21 pages, 12 figures, to appear in Eur. Phys. J.
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