Asymptotic behavior of non-autonomous Lamé systems with subcritical and critical mixed nonlinearities

This paper deals with the asymptotic behavior of solutions to a class of non-autonomous Lamé systems modeling the physical phenomenon of isotropic elasticity. The main feature of this model is that the nonlinearity can be decomposed into a subcritical part and a critical one. We first show that the system generates a non-autonomous dynamical system, and then prove that the system has a minimal universe pullback attractor. The upper-semicontinuity of these pullback attractors is also established as the perturbation parameter of the external force tends to zero. The quasi-stability ideas developed by Chueshov and Lasiecka (2010, 2008, 2015) are used to prove the pullback asymptotic compactness of the solutions in order to overcome the difficulty caused by the critical growthness of the nonlinearity.     

Some nonlinear problems in servomechanisms

Summary A mathematical model for a hydraulic servomechanism is constructed. It is shown that the model, in general, reduces to a nonlinear third-order equation of the formxx+(₁+xx+–₂ x=p(t). Under certain conditions imposed on the constants involved, it is proved that above equation possesses a periodic solution.     

Syntheses, Structures, and Nonlinear Optical Properties of Two Crown-like Heteroselenometallic Compounds [Et4N]4[(μ5-WSe4)(CuX)5(μ-X)2] (X = Cl, Br)

Two isostructural crown-like heteroselenometallic cluster compounds, [Et₄N]₄[(μ₅-WSe₄)(CuX)₅(μ-X)₂] (X = Cl 1, Br 2), were prepared from the reactions of [Et₄N]₂[WSe₄] with CuX and [Et₄N]X· xH₂O in the presence of 2-picoline and characterized by single-crystal diffraction analysis. The [(μ₅-WSe₄)(Cu-X)₅(μ-X)₂]⁴⁻ anions in the cluster compounds consists of five CuX fragments coordinated to the five edges of the tetrahedral [WSe₄]²⁻ moiety along with two bridging halides connected to each of the two pairs of the symmetric copper atoms, exhibiting a novel crown-like core structure. The nonlinear optical absorption and refraction of cluster compound 2 were determined to be α₂ = 6.15 × 10⁻¹⁰ m/W and n ₂ = 4.18 × 10⁻¹¹ esu, respectively.     

取代苯体系的二阶非线性光学性质:动力学李代数方法

提出了一种动力学李代数方法来研究取代苯体系的非线性光学性质. 对于给定的PPP模型(Pariser-Parr-Pople)哈密顿量, 生成了一个动力学李代数. 依据这些代数元构造出演化算子作为群参数的函数, 通过求解一组非线性微分方程能够得到这些群参数. 再按照统计力学中的密度算子公式给出取代苯分子体系偶极矩的统计平均值. 于是导出二阶极化率的表达式. 与其他量子力学计算结果比较, 表明这种动力学李代数方法在预言有机共轭分子的非线性光学性质上同样有用.     

金属团簇MoS4Cu4(SCN)2(NC5H5)6的皮秒光学非线性特性和光限幅特性研究

由于激光技术的快速发展，新型超快可调谐波长的高功率激光器更多的在民用和军事上应用。快速的非线性光学器件的需求越来越多。近10年，人们投入大量研究寻找超快光学非线性材料，同时寻找快速响应、高线性透过率和低非线性透过率以及高损伤阈值的光限幅材料，研究发现金属酞菁化合物、金属卟啉化合物以及富勒烯衍生物具有良好的光限幅效果和强的光学非线性特性。     

Mo(W)-Cu(Ag)-S簇合物的非线性光学性能研究

本文介绍了非线性光学性测试中的Ｚ－扫描方法。总结了Ｍｏ（Ｗ）－Ｃｕ（Ａｇ）－Ｓ簇合物的光限制效应、非线性吸收和非线性折射等光学性质。讨论了簇合物结构与光学性能的关系。     

Stable blow-up dynamics in the -critical and -supercritical generalized Hartree equation

We study stable blow-up dynamics in the generalized Hartree equation with radial symmetry, which is a Schrödinger-type equation with a nonlocal, convolution-type nonlinearity: First, we consider the -critical case in dimensions and obtain that a generic blow-up has a self-similar structure and exhibits not only the square root blowup rate , but also the log-log correction (via asymptotic analysis and functional fitting), thus, behaving similarly to the stable blow-up regime in the -critical nonlinear Schrödinger equation. In this setting, we also study blow-up profiles and show that generic blow-up solutions converge to the rescaled , a ground state solution of the elliptic equation . We also consider the -supercritical case in dimensions . We derive the profile equation for the self-similar blow-up and establish the existence and local uniqueness of its solutions. As in the NLS -supercritical regime, the profile equation exhibits branches of nonoscillating, polynomially decaying (multi-bump) solutions. A numerical scheme of putting constraints into solving the corresponding ordinary differential equation is applied during the process of finding the multi-bump solutions. Direct numerical simulation of solutions to the generalized Hartree equation by the dynamic rescaling method indicates that the is the profile for the stable blow-up. In this supercritical case, we obtain the blow-up rate without any correction. This blow-up happens at the focusing level , and thus, numerically observable (unlike the -critical case). In summary, we find that the results are similar to the behavior of stable self-similar blowup solutions in the corresponding settings for the nonlinear Schrödinger equation. Consequently, one may expect that the form of the nonlinearity in the Schrödinger-type equations is not essential in the stable formation of singularities.     

Nonexistence of global solutions of wave equations with weak time-dependent damping and combined nonlinearity

In our previous two works, we studied the blow-up and lifespan estimates for damped wave equations with a power nonlinearity of the solution or its derivative, with scattering damping independently. In this work, we are devoted to establishing a similar result for a combined nonlinearity. Comparing to the result of wave equation without damping, one can say that the scattering damping has no influence.