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 $\sigma=1$ or $-1$. In particular, for the integrable system with $\sigma=1$, 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 $\eta=\frac{x}{a(s)^{1/3}}$ with $s=3t;$ $\xi>0$ and $\alpha\geq0$ 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 $k_{\perp}=0$; 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