Incompressible and Compressible Limits of Coupled Systems of Nonlinear Schrödinger Equations

Recently, coupled systems of nonlinear Schrödinger equations have been used extensively to describe a double condensate, i.e. a binary mixture of Bose-Einstein condensates. In a double condensate, an interface and shock waves may occur due to large intraspecies and interspecies scattering lengths. To know the dynamics of an interface and assure the existence of shock waves in a double condensate, we study the incompressible and the compressible limits respectively of two coupled systems of nonlinear Schrödinger equations. The main idea of our arguments is to define a “H-functional” like a Lyapunov functional which can control the propagation of densities and linear momenta. Such an idea is different from the one using the standard Wigner transform to investigate the incompressible and the compressible limits of a single nonlinear Schrödinger equation. $\mathfrak{V}Recently, coupled systems of nonlinear Schr?dinger equations have been used extensively to describe a double condensate, i.e. a binary mixture of Bose-Einstein condensates. In a double condensate, an interface and shock waves may occur due to large intraspecies and interspecies scattering lengths. To know the dynamics of an interface and assure the existence of shock waves in a double condensate, we study the incompressible and the compressible limits respectively of two coupled systems of nonlinear Schr?dinger equations. The main idea of our arguments is to define a “H-functional” like a Lyapunov functional which can control the propagation of densities and linear momenta. Such an idea is different from the one using the standard Wigner transform to investigate the incompressible and the compressible limits of a single nonlinear Schr?dinger equation.      

Spinodal decomposition of an ABv model alloy: Patterns at unstable surfaces

We develop mean-field kinetic equations for a lattice gas model of a binary alloy with vacancies (ABv model) in which diffusion takes place by a vacancy mechanism. These equations are applied to the study of phase separation of finite portions of an unstable mixture immersed in a stable vapor. Due to a larger mobility of surface atoms, the most unstable modes of spinodal decomposition are localized at the vapor-mixture interface. Simulations show checkerboard-like structures at the surface or surface-directed spinodal waves. We determine the growth rates of bulk and surface modes by a linear stability analysis and deduce the relation between the parameters of the model and the structure and length scale of the surface patterns. The thickness of the surface patterns is related to the concentration fluctuations in the initial state. Received 28 October 1998     

Shock wave stability under the spinodal decomposition of binary mixtures

A diffusion equation for a binary mixture and spinodal decomposition in the case of phase separation are considered. It is shown that, if the binding force between polymer chain links is weak, the diffusion equation for a binary mixture allows for the reduction to the Burgers equation with “viscosity”; that is, the coexistence of rarefaction waves and shock density waves is a possibility. The effect of strong bonds between polymer chain links on the spinodal decomposition dynamics is studied. It is demonstrated that strong bonding may cause a multiflux wave system with alternate stability to arise when the viscosity varies.     

Shock Waves for a Discrete Velocity Gas Mixture

We introduce three new models for a binary mixture which have only 6+5, 8+5, and 12+5 velocities and study the properties of the first two. The models are plane and have five conservation laws as expected for a binary mixture in the plane case. We look for exact solutions corresponding to traveling waves, which turn out to have the properties of a structured shock wave, and study their properties. Particular attention is paid to the overshoots in the profiles of internal energy for the mixture and the two components.     

Interaction of Four Rarefaction Waves in the Bi-Symmetric Class of the Two-Dimensional Euler Equations

The global existence and structures of solutions to multi-dimensional unsteady compressible Euler equations are interesting and important open problems. In this paper, we construct global classical solutions to the interaction of four orthogonal planar rarefaction waves with two axes of symmetry for the Euler equations in two space dimensions, in the case where the initial rarefaction waves are large. The bi-symmetric initial data is a basic type of four-wave two-dimensional Riemann problems. The solutions in this case are continuous, bounded and self-similar, and we characterize how large the rarefaction waves must be. We use the methods of hodograph transformation, characteristic decomposition, and phase space analysis. We resolve binary interactions of simple waves in the process.     

旋转致密双星的引力波特征

研究了轨道和旋转效果到2.5阶后牛顿旋转致密双星拉格朗日动力学与引力波的关系, 分析了有序和混沌轨道的引力波特征.发现当加速度不考虑辐射项时, 有序双星系统辐射的引力波具有周期或拟周期的特征, 而混沌双星系统辐射的引力波却具有明显的混沌特征.当加速度含有辐射项贡献时, 双星必会出现并合现象.此时, 原保守有序双星系统需较长时间才能完成并合过程, 引力波形在双星并合前仍保留拟周期的基本特点;然而, 原保守混沌双星系统仅在较短时间内就会并合, 但因并合时间太短, 无法获取足够的动力学信息导致引力波形的特征不易分辨.     

The Doppler response to gravitational waves from a binary star source

The equations are developed for spacecraft Doppler detection of periodic gravitational waves from a single binary star source. Graphical examples are included to indicate the great variety of Doppler signals which can be generated by these systems.     

Thermodynamics of III–V solutions with n components

A method has been developed for the calculation of phase diagrams of III–V mixtures with more than three components. The equations for the chemical potentials, activity coefficients and the free energy of the mixture and for the liquidus- and iso-solidus-concentration lines are given. The parameters, needed for the calculation, are the temperature of fusion, heats of fusion, and the interaction parameters. The interaction parameters can be fitted to datas of the binary or quasi-binary phase diagrams, or can be calculated using the electronegativities, energies of sublimation and molar volumes of the constituent elements.Two approaches for a mixture with n components are given, which give in the ternary and binary case the known equations. The two possible cases of quaternary III–V solutions are discussed explicitly.     

Nonlinear equations for a slow magnetogasdynamic quasi-monochromatic wave in a medium with dispersion and dissipation

Evolutionary equations are obtained in the vicinity of a singularity for a slow wave with allowance for the quadratic nonlinearity, dispersion, and dissipation. It is demonstrated that the propagation of quasi-monochromatic waves in a current-conducting gas—liquid mixture is described by ordinary differential equations for the amplitudes in the aforementioned region.     

Optical Transfer Function of a Binary Markovian Mixture

A system of differential equations describing the optical transfer function (OTF) of a binary Markovian mixture is considered in a small-angle approximation. Its analytical solution has been found for the case where the mixture components have different extinction and scattering coefficients but identical scattering indicatrixes. The results of investigation show that the optical transfer function of the binary Markovian mixture depends strongly on the dimensions of nonuniformities and on the difference between the optical parameters of the mixture components, but it always markedly exceeds the optical transfer function of a uniform medium.     

Solid-liquid equilibria for quaternary solid solutions involving compound semiconductors in the regular solution approximation

Liquidus equations for solid-liquid equilibria in quaternary systems involving compound semiconductors are presented in both thermodynamic (model-independent) and regular solution forms. A unified treatment is offered for terminal (doubly-doped binary compound, singly-doped ternary solution) as well as continuous series of solid solutions (mixing on one or both sublattices) which is facilitated by the choice of binary compounds as components in the solid. With the exception of the quaternary solid solution with mixing on both sublattices (i.e. Al_uGa_1?uP_vAs_1?v), the liquidus equations are generalizations of previous results for ternary systems. In the case of mixing on both sublattices, only three of the four possible liquidus equations are independent on account of the equilibrium among the four compound components. The required binary compound activities in the quaternary solution are deduced from a statistical treatment of a regular mixture of four kinds of atoms subject to the site restrictions of the zinc-blende lattice. The calculation of quaternary phase diagrams from binary and ternary data is discussed. Finally, it is found for the case of mixing on both sublattices that previous results, based on the decomposition of the quaternary solid solution into a binary regular mixture of ternary solids, are consistent with the present analysis.     

Design of Si–SiO<Subscript>2</Subscript> phoxonic crystal having defect layer for simultaneous sensing of biodiesel in a binary mixture of diesel through optical and acoustic waves

The potentiality of a phoxonic crystal for sensing of biodiesel in a binary mixture of diesel and biodiesel is theoretically investigated. Using the transfer matrix method, the transmission of acoustic and optical waves through a periodic one-dimensional crystal of Si–SiO₂ layers is studied. A pass band is created in the band gap region by introducing a cavity in the considered one-dimensional crystal structure. This pass band can also be considered as a defect mode, and it is found that its position is highly dependent on mole concentration of binary mixture of biodiesel and diesel present in the cavity. The sensitivity of the sensor for a binary mixture of biodiesel and diesel in the cavity with various mole concentrations is estimated. Simulated results provide a valuable guidance for designing a phoxonic crystal sensor consisting of a defect layer.     

Dynamic Transition in a Two-Component Fluid in the Supercooled State

A mode-coupling model has been proposed to study the dynamics of a dense binary fluid. The present model includes a correct set of slow variables, which are consistent with the corresponding conservation laws. From the equations of fluctuating hydrodynamics for slow variables, a coupled set of nonlinear equations for various time-dependent correlations of density fluctuations are obtained. In the long-time limit, a self-consistent solution of these equations shows a dynamic transition, which occurs at a much higher density than predicted by the other existing models for the same systems. Our model brings the theoretical prediction for dynamic transition in better agreement with the simulation results for these systems. We have considered a hard sphere binary mixture to make a direct comparison of our results with the predictions of models studied in earlier works.     

A comparative study of the reflectivity of binary and ternary one-dimensional plasma photonic crystals for obliquely incident electromagnetic wave

In present paper the reflectivity of binary and ternary one-dimensional plasma photonic crystals for obliquely incident electromagnetic waves are studied. The reflectivity is derived using transfer matrix method and solving Maxwell’s equations by employing the continuity conditions on the electric fields and its derivatives. Our analysis shows that the perfect reflection bands get enhanced in the ternary 1D-PPCs compared to the binary 1D-PPCs. Also the ternary 1D-PPCs provide an additional degree of freedom to control the perfect reflection bands.     

Microscopic Hydrodynamic Modes in a Binary Hard Sphere Mixture

We derive analytic microscopic expressions for the shear viscosity, the speed of sound, and the decay rates of the hydrodynamic modes in a hard sphere binary gas mixture directly from the spectral properties of coupled Boltzmann equations. We show that the analytic expressions give good agreement with experimental viscosity data and to the results of light scattering experiments on noble gas binary mixtures.     

Sound waves in a liquid with polydisperse vapor–gas bubbles

A mathematical model is presented for the propagation of plane, spherical, and cylindrical sound waves in a liquid containing polydisperse vapor–gas bubbles with allowance for phase transitions. A system of integro-differential equations is constructed to describe perturbed motion of a two-phase mixture, and a dispersion relation is derived. An expression for equilibrium sound velocity is obtained for a gas–liquid or vapor–liquid mixture. The theoretical results agree well with the known experimental data. The dispersion curves obtained for the phase velocity and the attenuation coefficient in a mixture of water with vapor–gas bubbles are compared for various values of vapor concentration in the bubbles and various bubble distributions in size. The evolution of pressure pulses of plane and cylindrical waves is demonstrated for different values of the initial vapor concentration in bubbles. The calculated frequency dependence of the phase sound velocity in a mixture of water with vapor bubbles is compared with experimental data.     

Exact solutions for a semilinear hyperbolic system with general quadratic interactions

Summary The approximate conservation laws for a binary gas mixture with quadratic interactions can be written as a system of two semilinear hyperbolic equations. By using the method of characteristics the study of this system is reduced to the study of a system of nonlinear ordinary differential equations, for which we find several classes of exact solutions. Exact integrability conditions are found for ODE systems of generalized Lotka-Volterra type. Some situations are also considered, in which the characteristic method cannot be applied and exact solutions are obtained via an operator method.     

Nonlinear ultrasonic nature of organic liquid and organic liquid mixture

Based on Jacobson's molecular free length theory in liquids and the relationship between ultrasonic velocity and the molecular free length in organic liquids, this paper deduces the equations for pressure coefficient and temperature coefficient of ultrasonic velocity and nonlinear acoustic parameter B/A in both of organic liquid and organic liquid binary mixtures. These nonlinear acoustic parameters are evaluated against the measured results and data from other sources. The equations reveal the connections between the nonlinear acoustic parameters and some internal structural of the medium or mixtures e.g. the sizes of molecule, several thermodynamic physical parameters and outside status e.g. condition of pressure and temperature of the liquid or liquid mixture. With the equations the nonlinear acoustic parameter B/A of organic liquid binary mixtures, which is impossible to know without the nonlinear acoustic parameter B/A of the tow components before, can be calculated based on the structural and physical parameters of organic liquid and organic liquid binary mixtures.     

Hydrodynamics of binary fluid phase segregation

Starting with the Vlasov-Boltzmann equation for a binary fluid mixture, we derive an equation for the velocity field u when the system is segregated into two phases (at low temperatures) with a sharp interface between them. u satisfies the incompressible Navier-Stokes equations together with a jump boundary condition for the pressure across the interface which, in turn, moves with a velocity given by the normal component of u. Numerical simulations of the Vlasov-Boltzmann equations for shear flows parallel and perpendicular to the interface in a phase segregated mixture support this analysis. We expect similar behavior in real fluid mixtures.     

统一性溶液理论及其应用

本文基于正规溶液模型和胞腔溶液模型构作了一个称之为统一性的溶液理论，并以此建立了相应的超额Ｇｉｂｂｓ自由能和活度系度数方程。另外，还应用该模型的活度系数方程计算了一些具有代表性体系的汽液平衡，取得很好的结果。