Simple model of skin formation caused by solvent evaporation in polymer solutions

A simple model is proposed for the skin formation in the evaporation process of a polymer solution at a free surface. In this model the skin is regarded as a gel phase formed near the free surface, and the dynamics is described by a diffusion equation for the polymer concentration with moving boundaries. The equation is solved both analytically and numerically. It is shown that the skin phase appears when the evaporation rate is high or when the initial polymer concentration is high. An analytical expression is given for the criterion for the skin phase to be formed.     

Evolution of the structure factor in a hyperbolic model of spinodal decomposition

We consider the modification of the Cahn-Hilliard equation when a time delay process through a memory function is taken into account. The memory effects are seen to affect the dynamics of phase transition at short times. The process of fast spinodal decomposition associated with a conserved order parameter - concentration is studied numerically. Details of a semi-implicit numerical scheme used to simulate the kinetics of spinodal decomposition and evolution of the structure factor are discussed. Analysis of the modeled structure factor predicted by a hyperbolic model of spinodal decomposition is presented in comparison with the parabolic model of Cahn and Hilliard. It is shown that during initial periods of decomposition the structure factor exhibits wave behavior. Analytical treatments explain such behavior by existence of damped oscillations in structure factor at earliest stages of phase separation and at large values of the wave-number. These oscillations disappear gradually in time and the hyperbolic evolution approaches the pure dissipative parabolic evolution of spinodal decomposition.     

立方非线性Schr?dinger方程的动力学性质研究及其解模式的漂移

采用辛算法数值求解一维立方非线性Schr?dinger方程，研究了随着非线性参数的变化立方非线性Schr?dinger方程的动力学性质和解的模式的漂移.数值结果表明，随着非线性参数的增加解模式的漂移速度越来越快. 关键词： 动力学性质 相轨线 解模式的漂移 辛算法     

基于局部加密纯无网格法非线性Cahn-Hilliard方程的模拟

为数值求解描述不同物质间相位分离现象的高阶非线性Cahn-Hilliard(C-H)方程,发展了一种基于局部加密纯无网格有限点集法(local refinement finite pointset method,LR-FPM).其构造过程为:1)将C-H方程中四阶导数降阶为两个二阶导数,连续应用基于Taylor展开和加权最小二乘法的FPM离散空间导数;2)对区域进行局部加密和采用五次样条核函数以提高数值精度;3)局部线性方程组求解中准确施加含高阶导数Neumann边值条件.随后,运用LR-FPM求解有解析解的一维/二维C-H方程,分析粒子均匀分布/非均匀分布以及局部粒子加密情况的误差和收敛阶,展示了LR-FPM较网格类算法在非均匀布点情况下的优点.最后,采用LR-FPM对无解析解的一维/二维C-H方程进行了数值预测,并与有限差分结果相比较.数值结果表明,LR-FPM方法具有较高的数值精度和收敛阶,比有限差分法更易数值实现,能够准确展现不同类型材料间相位分离非线性扩散现象随时间的演化过程.     

Eigenvalues of the Zakharov-Shabat scattering problem for two separated sech-shaped pulses

It is shown that the initial condition of two separated sech-shaped in-phase pulses in the nonlinear Schrödinger equation, may give rise to not only stationary solitons, but also symmetrically separating solitons, provided the initial distance of separation is large enough. The critical distance between the pulses for which a separating soliton pair can be found for certain amplitudes is derived using a variational approach.     

A method of investigation of the evolution of a nonlinear system driven by pulsed noise

A numerical method is proposed for determining the evolution of nonlinear systems subjected to noise. The method is based on a recurrence equation for the probability density which has been obtained analytically due to the choice of noise in the form of discrete series of random pulses. The method is applied to a dynamical system which describes the motion of a particle in a plane-wave field. The evolution of the probability density in phase and energy space is obtained. It is shown that because of noise effects, the region in phase space where particles can be found rapidly reaches the separatrix and then spreads over the phase space, mainly along the separatrix. In the energy spectrum a new peak appears at the separatrix's energy. This peak grows in time, while the main peak corresponding to the initial energy drops in time and shifts to lower energy. The moments of motion were analyzed. The character of their evolution indicates a high rate of chaotization. The growth of the fraction of energetic particles is very rapid (exponential at the beginning), whereas the mean energy grows linearly.     

Nonlinear theory of ionic sound waves in a hot quantum-degenerate electron-positron-ion plasma

A collisionless nonmagnetized e-p-i plasma consisting of quantum-degenerate gases of ions, electrons, and positrons at nonzero temperatures is considered. The dispersion equation for isothermal ionic sound waves is derived and analyzed, and an exact expression is obtained for the linear velocity of ionic sound. Analysis of the dispersion equation has made it possible to determine the ranges of parameters in which nonlinear solutions in the form of solitons should be sought. A nonlinear theory of isothermal ionic sound waves is developed and used for obtaining and analyzing the exact solution to the system of initial equations. Analysis has been carried out by the method of the Bernoulli pseudopotential. The ranges of phase velocities of periodic ionic sound waves and soliton velocities are determined. It is shown that in the plasma under investigation, these ranges do not overlap and that the soliton velocity cannot be lower than the linear velocity of ionic sound. The profiles of physical quantities in a periodic wave and in a soliton are constructed, as well as the dependences of the velocity of sound and the critical velocity on the ionic concentration in the plasma. It is shown that these velocities increase with the ion concentration.     

Stability of liquid film falling down a vertical non-uniformly heated wall

The main object in this paper is to study the stability of a viscous film flowing down a vertical non-uniformly heated wall under gravity. The wall temperature is assumed linearly distributed along the wall and the free surface is taken to be adiabatic. A long wave perturbation method is used to derive the nonlinear evolution equation for the falling film. Using the method of multiple scale, the nonlinear stability analysis is studied for travelling wave solution of the evolution equation. The complex Ginzburg-Landau equation is determined to discuss the bifurcation analysis of the evolution equation. The results indicate that the supercritical unstable region increases and the subcritical stable region decreases with the increase of Peclet number. It has been also shown that the spatial uniform solution corresponding to the sideband disturbance may be stable in the unstable region.     

非线性动态特性的逆推演方法及其在干涉术中的应用

提出了研究对象非线性动态特性的逆推演新方法,它的特点是从已知对象的非微分方程解析解导出该对象可能的微分方程,从而开拓它的解域或解空间,或者说恢复丢失的解。同时讨论了当对象为法布里－珀罗干涉（F－P）或迈克耳孙干涉时该方法的应用过程,并且对法布里－珀罗定义新的不透过系数（u＝1／τ）,从而给出更简洁和易分析的对象动态特性微分议程,以及由此导出测量法布里－珀罗干涉相位的过程,值得注意的是由新方法导出的微分议程揭示对象更一般的时空演化特征。而且在现有的经验和知识基础上可以进一步唯象地拓展至其他可能的非线性形式,从而使对象的表达方式更接近它的实际情况。     

Effect of Bath Concentration on Coagulation Kinetics at the Early Stage during Wet Spinning of PAN Copolymer Nascent Fibers

A mathematical model was proposed to simulate the effect of bath concentration on coagulation kinetics at the early stage of wet spinning for the poly(acrylonitrile-co-vinyl acetate)/dimethylsulfoxide (DMSO)/water system. The dependence of critical precipitation time, components concentration distribution, and the radius of nascent fibers on the concentration of DMSO in the DMSO/water coagulation bath were estimated by solving the model equation numerically. The experimental results indicated the model was suitable to simulate the dynamic features of the early stage of the coagulation process. The critical precipitation time was found to increase with bath concentration. The mode of phase separation was changed from instantaneous demixing to delayed demixing as DMSO bath concentration increased. The simulation results showed that bath concentration influenced the phase separation path which determined the polymer concentration distribution in the spinning solution. As a result, nascent fibers with different structures would form in wet spinning and a radial homogeneous structure would be obtained when the DMSO bath concentration increased to some extent.     

一类扰动非线性发展方程的类孤子同伦近似解析解

采用一个简单而有效的技巧,研究了一类扰动非线性发展方程.首先,引入一个相应典型方程的孤子近似解.然后,利用同伦映射方法得到了原扰动非线性发展方程的近似解. 关键词： 孤子 扰动发展方程 同伦映射     

Relaxation processes in phonon–impuriton system and effective thermal conductivity of superfluid He–He solutions

By the method of stationary thermal flow, the measurements of effective thermal conductivity in the superfluid solution ³He–⁴He with initial molar concentration 9.8% ³He in the temperature range 0.1–0.5 K have been carried out. At temperatures below 0.23 K, where the phase separation of the mixture occurs, the measurements were carried out in the lower superfluid phase with changing concentration following the phase diagram with decreasing the temperature. Experimental data were compared with the prediction of the kinetic theory of the phonon–impuriton system of a solution. The analysis allows us to establish the relaxation time hierarchy in such a system and to provide the quantitative estimation of the times of impuriton–impuriton and phonon–impuriton relaxation. It is shown that a rather good agreement between experimental data and theory can be obtained by the use of one fit parameter.     

The modify unstable nonlinear Schrödinger dynamical equation and its optical soliton solutions

In this research, we work on a specific class of nonlinear evolution equation which is the modify unstable nonlinear Schrödinger equation. This equation is used to describe a time evolution of disturbances in unstable media. Various solutions have been obtained. The results deduced are of varied types and include bright solution, dark solution, rational dark-bright solution, as well as cnoidal solutions. These solutions might be useful in engineering fields. Some conditions for the stability of these solutions are presented. The method used here is understandable and very powerful for solving the nonlinear problems.     

Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The data contain errors (observation and background errors), hence there will be errors in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can often be approximated by the inverse Hessian of the cost functional. Here we focus on highly nonlinear dynamics, in which case this approximation may not be valid. The equation relating the optimal solution error and the errors of the input data is used to construct an approximation of the optimal solution error covariance. Two new methods for computing this covariance are presented: the fully nonlinear ensemble method with sampling error compensation and the ‘effective inverse Hessian’ method. The second method relies on the efficient computation of the inverse Hessian by the quasi-Newton BFGS method with preconditioning. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term.     

Elimination of fast mode in hydrodynamics of superfluid turbulence

Using the multi-time scales method we fulfill procedure of a separation of a fast and slow processes in the equations of HST. It is shown that slow stage of the evolution of transient heat load of moderate intensity obeys to the well-known nonlinear heat conductivity equation.     

Dynamics of large-amplitude solitons

The interaction and generation of solitons in nonlinear integrable systems which allow the existence of a soliton of limiting amplitude are considered. The integrable system considered is the Gardner equation, which includes the Korteweg-de Vries equation (for quadratic nonlinearity) and the modified Korteweg-de Vries equation (for cubic nonlinearity) as special cases. A two-soliton solution of the Gardner equation is derived, and a criterion, which distinguishes between different scenarios for the interaction of two solitons, is determined. The evolution of an initial pulsed disturbance is considered. It is shown, in particular, that solitons of opposite polarity appear during such evolution on the crest of a limiting soliton. Zh. éksp. Teor. Fiz. 116, 318–335 (July 1999)     

Theory of microphase separation in charged crosslinked polymer blends with arbitrary charge distribution

In this paper, we are interested in the phase behavior and scattering properties of charged crosslinked polymer blends in solution. The system undergoes a microphase separation, below some critical temperature. To study such a transition, use is made of the standard de Gennes theory based on an analogy with a dielectric medium. This analogy is extended to include the effects of the initial composition fluctuations in order to improve its agreement with experimental data in the small wave vector range. The excluded-volume interactions are explicitly introduced through the blob model. The charge effects on the phase behavior are examined, for any charge distribution of polyions and for any salt concentration. This completes a previous study which was concerned with the situation where only one species is charged. The early kinetics of microphase separation is discussed, and the charges contribution to the growth rate is also evaluated.     

Phase separated composite films of liquid crystals

Phase separation of liquid crystals from a solution with polymers has long been studied and used to prepare polymer stabilized and polymer dispersed structures. They are formed by spatially isotropic phase separation. A new mode, in which the phase separation proceeds anisotropically, has recently been discovered. Known as phase separated composite films (PSCOF), the resultant structures are made of adjacent parallel layers of liquid crystal and solidified polymer. PSCOFs have been made with nematic, ferroelectric (FLC), and antiferroelectric (AFLC) liquid crystals. Liquid crystals in PSCOFs exhibit electro-optical properties not observed in devices prepared by conventional methods, polymer dispersion, or polymer stabilization methods. Devices incorporating FLCs possess grey scale and switch 100 times faster at low fields than conventional surface stabilized devices. This method makes it possible to prepare very flexible devices and devices with liquid crystal film thickness comparable to optical wavelengths with great ease.     

The evolution equation for the radius of a premixed flame ball in fractional diffusive media

The evolution equation for the radius of an isolated premixed flame ball is derived in the framework of a new method that strongly simplifies previous ones and highlights that they are based on Gaussian modelling of diffusion. The main idea is to split the flame ball in two components: the inner kernel, which is driven by a Poisson-type equation with a general polynomial forcing term, and the outer part, which is driven by a generalized diffusion process valid for fractional diffusive media. The evolution equation for the radius of the flame ball is finally determined as the evolution equation for the interface that matches the solution of the inner spherical kernel and the solution of the outer diffusive part and it emerges to be a nonlinear fractional differential equation. The effects of fractional diffusion on stability of solution are also picked out.     

Generation and propagation of pulse trains with ultrashort pulse separation

We investigate the nonlinear Schrödinger equation with variable coefficients by employing perturbation method. The analysis solution of the harmonic form is presented. The solution is one of forms to describe pulse trains with ultrashort pulse separation, which is about two orders of magnitude shorter than one of sech-type solitons considered before. And we could systematically adjust the perturbation parameter to obtain different pulse separation. As an example, we consider a nonlinear dispersive system with spatial parameter variations, and the results show that, the pulse train with ultrashort pulse separation presented by analysis solution may keep its shape even if the velocity is changed. The stability of the solution is discussed numerically, and the results reveal that the finite initial perturbations, such as white noise could not influence the main character of the solution. In addition, the stability of the solution is also discussed under more general conditions.