10,425 research outputs found
Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations
In this article, we study the self-similar solutions of the 2-component
Camassa-Holm equations% \begin{equation} \left\{ \begin{array} [c]{c}%
\rho_{t}+u\rho_{x}+\rho u_{x}=0
m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right. \end{equation}
with \begin{equation} m=u-\alpha^{2}u_{xx}. \end{equation} By the separation
method, we can obtain a class of blowup or global solutions for or
. In particular, for the integrable system with , we have the
global solutions:% \begin{equation} \left\{ \begin{array} [c]{c}%
\rho(t,x)=\left\{ \begin{array} [c]{c}% \frac{f\left( \eta\right)
}{a(3t)^{1/3}},\text{ for }\eta^{2}<\frac {\alpha^{2}}{\xi}
0,\text{ for }\eta^{2}\geq\frac{\alpha^{2}}{\xi}% \end{array} \right.
,u(t,x)=\frac{\overset{\cdot}{a}(3t)}{a(3t)}x
\overset{\cdot\cdot}{a}(s)-\frac{\xi}{3a(s)^{1/3}}=0,\text{ }a(0)=a_{0}%
>0,\text{ }\overset{\cdot}{a}(0)=a_{1}
f(\eta)=\xi\sqrt{-\frac{1}{\xi}\eta^{2}+\left( \frac{\alpha}{\xi}\right)
^{2}}% \end{array} \right. \end{equation}
where with and are
arbitrary constants.\newline Our analytical solutions could provide concrete
examples for testing the validation and stabilities of numerical methods for
the systems.Comment: 5 more figures can be found in the corresponding journal paper (J.
Math. Phys. 51, 093524 (2010) ). Key Words: 2-Component Camassa-Holm
Equations, Shallow Water System, Analytical Solutions, Blowup, Global,
Self-Similar, Separation Method, Construction of Solutions, Moving Boundar
Bubble Raft Model for a Paraboloidal Crystal
We investigate crystalline order on a two-dimensional paraboloid of
revolution by assembling a single layer of millimeter-sized soap bubbles on the
surface of a rotating liquid, thus extending the classic work of Bragg and Nye
on planar soap bubble rafts. Topological constraints require crystalline
configurations to contain a certain minimum number of topological defects such
as disclinations or grain boundary scars whose structure is analyzed as a
function of the aspect ratio of the paraboloid. We find the defect structure to
agree with theoretical predictions and propose a mechanism for scar nucleation
in the presence of large Gaussian curvature.Comment: 4 pages, 4 figure
Dynamics of a liquid dielectric attracted by a cylindrical capacitor
The dynamics of a liquid dielectric attracted by a vertical cylindrical
capacitor is studied. Contrary to what might be expected from the standard
calculation of the force exerted by the capacitor, the motion of the dielectric
is different depending on whether the charge or the voltage of the capacitor is
held constant. The problem turns out to be an unconventional example of
dynamics of a system with variable mass, whose velocity can, in certain
circumstances, suffer abrupt changes. Under the hypothesis that the voltage
remains constant the motion is described in qualitative and quantitative
details, and a very brief qualitative discussion is made of the constant charge
case.Comment: To appear in European Journal of Physic
Diffusive transport in networks built of containers and tubes
We developed analytical and numerical methods to study a transport of
non-interacting particles in large networks consisting of M d-dimensional
containers C_1,...,C_M with radii R_i linked together by tubes of length l_{ij}
and radii a_{ij} where i,j=1,2,...,M. Tubes may join directly with each other
forming junctions. It is possible that some links are absent. Instead of
solving the diffusion equation for the full problem we formulated an approach
that is computationally more efficient. We derived a set of rate equations that
govern the time dependence of the number of particles in each container
N_1(t),N_2(t),...,N_M(t). In such a way the complicated transport problem is
reduced to a set of M first order integro-differential equations in time, which
can be solved efficiently by the algorithm presented here. The workings of the
method have been demonstrated on a couple of examples: networks involving
three, four and seven containers, and one network with a three-point junction.
Already simple networks with relatively few containers exhibit interesting
transport behavior. For example, we showed that it is possible to adjust the
geometry of the networks so that the particle concentration varies in time in a
wave-like manner. Such behavior deviates from simple exponential growth and
decay occurring in the two container system.Comment: 21 pages, 18 figures, REVTEX4; new figure added, reduced emphasis on
graph theory, additional discussion added (computational cost, one
dimensional tubes
Amplitude Modulation and Relaxation-Oscillation of Counterpropagating Rolls within a Broken-Symmetry Laser-Induced Electroconvection Strip
We report a liquid-crystal pattern-formation experiment in which we break the
lateral (translational) symmetry of a nematic medium with a laser-induced
thermal gradient. The work is motivated by an improved measurement (reported
here) of the temperature dependence of the electroconvection threshold voltage
in planar-nematic 4-methoxybenzylidene-4-butylaniline (MBBA). In contrast with
other broken-symmetry-pattern studies that report a uniform drift, we observe a
strip of counterpropagating rolls that collide at a sink point, and a strong
temporally periodic amplitude modulation within a width of 3-4 rolls about the
sink point. The time dependence of the amplitude at a fixed position is
periodic but displays a nonsinusoidal relaxation-oscillation profile. After
reporting experimental results based on spacetime contours and wavenumber
profiles, along with a measurement of the change in the drift frequency with
applied voltage at a fixed control parameter, we propose some potential
guidelines for a theoretical model based on saddle-point solutions for
Eckhaus-unstable states and coupled complex Ginzburg-Landau equations.
Published in PRE 73, 036317 (2006).Comment: Published in Physical Review E in March 200
Principle of Maximum Entropy Applied to Rayleigh-B\'enard Convection
A statistical-mechanical investigation is performed on Rayleigh-B\'enard
convection of a dilute classical gas starting from the Boltzmann equation. We
first present a microscopic derivation of basic hydrodynamic equations and an
expression of entropy appropriate for the convection. This includes an
alternative justification for the Oberbeck-Boussinesq approximation. We then
calculate entropy change through the convective transition choosing mechanical
quantities as independent variables. Above the critical Rayleigh number, the
system is found to evolve from the heat-conducting uniform state towards the
convective roll state with monotonic increase of entropy on the average. Thus,
the principle of maximum entropy proposed for nonequilibrium steady states in a
preceding paper is indeed obeyed in this prototype example. The principle also
provides a natural explanation for the enhancement of the Nusselt number in
convection.Comment: 13 pages, 4 figures; typos corrected; Eq. (66a) corrected to remove a
double counting for ; Figs. 1-4 replace
Stellar Hydrodynamics in Radiative Regions
We present an analysis of the response of a radiative region to waves
generated by a convective region of the star; this wave treatment of the
classical problem of ``overshooting'' gives extra mixing relative to the
treatment traditionally used in stellar evolutionary codes. The interface
between convectively stable and unstable regions is dynamic and nonspherical,
so that the nonturbulent material is driven into motion, even in the absence of
``penetrative overshoot.'' These motions may be described by the theory of
nonspherical stellar pulsations, and are related to motion measured by
helioseismology. Multi-dimensional numerical simulations of convective flow
show puzzling features which we explain by this simplified physical model.
Gravity waves generated at the interface are dissipated, resulting in slow
circulation and mixing seen outside the formal convection zone. The approach
may be extended to deal with rotation and composition gradients. Tests of this
description in the stellar evolution code TYCHO produce carbon stars on the
asymptotic giant branch (AGB), an isochrone age for the Hyades and three young
clusters with lithium depletion ages from brown dwarfs, and lithium and
beryllium depletion consistent with observations of the Hyades and Pleiades,
all without tuning parameters. The insight into the different contributions of
rotational and hydrodynamic mixing processes could have important implications
for realistic simulation of supernovae and other questions in stellar
evolution.Comment: 27 pages, 5 figures, accepted to the Astrophysical Journa
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