若干高维可积和不可积模型的严格解

可积和不可积模型可以描述自然科学中的诸多现象, 寻找高维非线性模型的严格解已成为可积系统的一个重要研究内容. 结合达布变换法和多线性分离变量法, 可以得到多个(2+1)维非线性模型包含任意函数的严格解, 通过选取不同的任意函数, 构造这些非线性模型新的相互激发模式. 进一步推广了形变映射理论, 建立了变系数 场和sine-Gordon以及双sine-Gordon场的形变映射关系, 从而得到高维不可积模型包含任意函数的新严格解. 对任意函数的不同选择, 构造了sine-Gordon和双sine-Gordon可积模型丰富的局域解和周期解, 如多solitoff解及其周期波推广、周期形变的蛇形孤波解以及变模的拟周期解等.     

Engineered Anisotropic Fluids of Rare-Earth Nanomaterials

The self-assembly of inorganic nanoparticles into well-ordered structures in the absence of solvents is generally hindered by van der Waals forces, leading to random aggregates between them. To address the problem, we functionalized rigid rare-earth (RE) nanoparticles with a layer of flexible polymers by electrostatic complexation. Consequently, an ordered and solvent-free liquid crystal (LC) state of RE nanoparticles was realized. The RE nanomaterials including nanospheres, nanorods, nanodiscs, microprisms, and nanowires all show a typical nematic LC phase with one-dimensional orientational order, while their microstructures strongly depend on the particles’ shape and size. Interestingly, the solvent-free thermotropic LCs possess an extremely wide temperature range from −40 °C to 200 °C. The intrinsic ordering and fluidity endow anisotropic luminescence properties in the system of shearing-aligned RE LCs, offering potential applications in anisotropic optical micro-devices.     

Self-Assembly Anisotropic Magnetic Nanowire Films Induced by External Magnetic Field

Self-assembly generated materials induced by an external magnetic field have attracted considerable interest following the development of nanodevices. However, the fabrication of macroscopic and anisotropic magnetic films at the nanoscale remains a challenge. Here, anisotropic magnetic films are successfully prepared using a solution-based nanowire assembly strategy under a magnetic field. The assembly process is manipulated by changing the thickness of silica shell coated on the surface of magnetic nanowires. The anisotropic magnetic films show highly anisotropic magnetization under different angles of magnetic field and better magnetization properties than that of disordered magnetic films. The well-defined nanowire arrays enable magnetization anisotropic property which may be useful in the magnetic energy conversion technologies and biomedical sciences which lie far beyond those achievable with traditional magnetic materials.     

Effects of aromatic regularity on the structure and conductivity of polyimide‐poly(ethylene glycol) materials doped with ionic liquid

An understanding of the structure and properties of polymer electrolyte systems can be crucial to a variety of different applications. The current work performs a study of the composition, structure and properties of poly(ethylene glycol) (PEG)‐aromatic polyimide systems incorporating ionic liquids that are relevant to several applications especially fuel cell membranes. Composition was varied through using different aromatic dianhydrides, aromatic diamines and in some cases synthesis solvent. Properties were characterized using Fourier transform infrared spectroscopy, thermal gravimetric analysis, differential scanning calorimetry, small‐angle x‐ray scattering, electrochemical impedance spectroscopy and cyclic voltammetry. By varying solvent, aromatic regularity and expected rigidity can be tuned, impacting average conductivity by 30%. Varying the aromatic diamine can influence the length scale and amount of aromatic regularity, which can ultimately affect the conductivity by a factor of four. The maximum conductivity reached was 83 mS/cm at 80 °C and 70 %RH. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 509–521     

Beyond microstructures: Using the Kerr Effect to characterize the macrostructures of synthetic polymers

The macrostructures of synthetic polymers are essentially the complete molecular chain architectures, including the types and amounts of constituent short‐range microstructures, such as the regio‐ and stereosequences of the inserted monomers, the amounts and sequences of monomers found in co‐, ter‐, and tetra‐polymers, branching, inadvertent, and otherwise, etc. Currently, the best method for characterizing polymer microstructures uses high field, high resolution ¹³C‐nuclear magnetic resonance (NMR) spectroscopy observed in solution. However, even ¹³C‐NMR is incapable of determining the locations or positions of resident polymer microstructures, which are required to elucidate their complete macrostructures. The sequences of amino acid residues in proteins, or their primary structures, cannot be characterized by NMR or other short‐range spectroscopic methods, but only by decoding the DNA used in their syntheses or, if available, X‐ray analysis of their single crystals. Similarly, there are currently no experimental means to determine the sequences or locations of constituent microstructures along the chains of synthetic macromolecules. Thus, we are presently unable to determine their macrostructures. As protein tertiary and quaternary structures and their resulting ultimate functions are determined by their primary sequence of amino acids, so too are the behaviors and properties of synthetic polymers critically dependent on their macrostructures. We seek to raise the consciousness of both synthetic and physical polymer scientists and engineers to the importance of characterizing polymer macrostructures when attempting to develop structure–property relations. To help achieve this task, we suggest using the electrical birefringence or Kerr effects observed in their dilute solutions. The molar Kerr constants of polymer solutes contributing to the birefringence of their solutions, under the application of a strong electric field, are highly sensitive to both the types and locations of their constituent microstructures. As a consequence, we may begin to characterize the macrostructures of synthetic polymers by means of the Kerr effect. To simplify implementation of the Kerr effect to characterize polymer macrostructures, we suggest that NMR first be used to determine the types and amounts of constituent microstructures present. Subsequent comparison of observed Kerr effects with those predicted for different microstructural locations along the polymer chains can then be used to identify the most likely macrostructures. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 155–166     

New implicitly solvable potential produced by second order shape invariance

The procedure proposed recently by Bougie et al. (2010) to study the general form of shape invariant potentials in one-dimensional Supersymmetric Quantum Mechanics (SUSY QM) is generalized to the case of Higher Order SUSY QM with supercharges of second order in momentum. A new shape invariant potential is constructed by this method. It is singular at the origin, it grows at infinity, and its spectrum depends on the choice of connection conditions in the singular point. The corresponding Schrödinger equation is solved explicitly: the wave functions are constructed analytically, and the energy spectrum is defined implicitly via the transcendental equation which involves Confluent Hypergeometric functions.     

On the accuracy of a nonlinear finite volume method for the solution of diffusion problems using different interpolations strategies

In this paper, we consider a nonlinear finite volume method to solve the steady‐state diffusion equation in nonhomogeneous and non‐isotropic media. The method is nonlinear even if the original problem is linear. In its original form, the scheme is monotone, because the coefficient matrix is monotone under certain assumptions and, as a consequence, whenever the analytic operator demands, it preserves the positivity of numerical solutions. On the other hand, the scheme is unable to reproduce piecewise linear solutions exactly. In order to recover this interesting feature, we use two different interpolation strategies. In this case, even though we are unable to prove monotonicity, we show some numerical evidences that the combined method has an improved behavior, producing second order accurate solutions, even for nonhomogeneous and strongly anisotropic media. Copyright © 2013 John Wiley & Sons, Ltd.     

Nonlinear corrector for Reynolds-averaged Navier-Stokes equations

The scope of this paper is to present a nonlinear error estimation and correction for Navier-Stokes and Reynolds-averaged Navier-Stokes equations. This nonlinear corrector enables better solution or functional output predictions at fixed mesh complexity and can be considered in a mesh adaptation process. After solving the problem at hand, a corrected solution is obtained by solving again the problem with an added source term. This source term is deduced from the evaluation of the residual of the numerical solution interpolated on the h/2 mesh. To avoid the generation of the h/2 mesh (which is prohibitive for realistic applications), the residual at each vertex is computed by local refinement only in the neighborhood of the considered vertex. One of the main feature of this approach is that it automatically takes into account all the properties of the considered numerical method. The numerical examples point out that it successfully improves solution predictions and yields a sharp estimate of the numerical error. Moreover, we demonstrate the superiority of the nonlinear corrector with respect to linear corrector that can be found in the literature.     

Structural transformation from shish‐kebab crystals to micro‐fibrils through hot stretching process of gel‐spun ultra‐high molecular weight polyethylene fibers with high concentration solution

Structural evolution of gel‐spun ultra‐high molecular weight polyethylene fibers with high concentration solution via hot stretching process was investigated by in situ small‐angle X‐ray scattering, in situ wide‐angle X‐ray diffraction measurements, scanning electron microscopy, and differential scanning calorimetry. With the increase of stretching strain, the long period continuously increases at relative lower stretching temperature, while it first increases and then decreases rapidly at relative higher stretching temperature. The kebab thickness almost keeps constant during the whole hot‐stretching process and the kebab diameter continually decreases for all stretching temperatures. Moreover, the length of shish decreases slightly and the shish quantity increases although there is almost no change in the diameter of shish crystals during the hot stretching process. The degree of crystal orientation at different temperatures is as high as above 0.9 during the whole stretching process. These results indicate that the shish‐kebab crystals in ultra‐high molecular weight polyethylene fibers can transform continuously into the micro‐fibril structure composed mostly of shish crystals through the hot stretching process. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2018 , 56, 225–238     

A higher resolution edge‐based finite volume method for the simulation of the oil–water displacement in heterogeneous and anisotropic porous media using a modified IMPES method

In this article, we present a higher‐order finite volume method with a ‘Modified Implicit Pressure Explicit Saturation’ (MIMPES) formulation to model the 2D incompressible and immiscible two‐phase flow of oil and water in heterogeneous and anisotropic porous media. We used a median‐dual vertex‐centered finite volume method with an edge‐based data structure to discretize both, the elliptic pressure and the hyperbolic saturation equations. In the classical IMPES approach, first, the pressure equation is solved implicitly from an initial saturation distribution; then, the velocity field is computed explicitly from the pressure field, and finally, the saturation equation is solved explicitly. This saturation field is then used to re‐compute the pressure field, and the process follows until the end of the simulation is reached. Because of the explicit solution of the saturation equation, severe time restrictions are imposed on the simulation. In order to circumvent this problem, an edge‐based implementation of the MIMPES method of Hurtado and co‐workers was developed. In the MIMPES approach, the pressure equation is solved, and the velocity field is computed less frequently than the saturation field, using the fact that, usually, the velocity field varies slowly throughout the simulation. The solution of the pressure equation is performed using a modification of Crumpton's two‐step approach, which was designed to handle material discontinuity properly. The saturation equation is solved explicitly using an edge‐based implementation of a modified second‐order monotonic upstream scheme for conservation laws type method. Some examples are presented in order to validate the proposed formulation. Our results match quite well with others found in literature. Copyright © 2016 John Wiley & Sons, Ltd.