A note on the necessary conditions for the algebraic Riccati equation

We derive some useful and easily computable necessary conditionsfor the existence of a positive semi-definite solution to thealgebraic Ricatti equation (ARE). A motivating example is givento highlight the usefulness of the conditions for controllerand observer designs for nonlinear systems. Further, an upperbound on the trace of the solution to the ARE is also derived.     

初-边值问题驻波解的渐近稳定性

本文考虑一带有人工粘性的二维定常等熵无旋平面流方程组的初-边值问题. 在一定的假设下,我们证明其驻波解是渐近稳定的.     

The Convergence and Stability of Splitting Finite-Difference Schemes for Nonlinear Evolutionary Equations

We consider a splitting finite-difference scheme for an initial-boundary value problem for a two-dimensional nonlinear evolutionary equation. The problem is split into nonlinear and linear parts. The linear part is also split into locally one-dimensional equations. We prove the convergence and stability of the scheme in L ₂ and C norms. Printed in Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 413–434, July–September, 2005.     

Optical bistability in an all-optical fiber nonlinear fabry-perot resonator with linear absorption

An all-optical fiber nonlinear Fabry-Perot resonator is proposed. The steady-state model for describing this resonator is developed. The transmission and refection bistabilities are shown numerically. Our results show that the bistability in this system is similar to that in a normal nonlinear Fabry-Perot etalon, except that the characteristics of the bistability can be changed not only by the cavity length but also by the phase shifts of the fiber loop mirrors.     

KDP晶体三倍频实验和分析

报道了ＫＤＰ晶体中的三倍频产生的实验研究，计算了比较了直接三倍频过程和级联过程对ＫＤＰ晶体三阶有效非线性系数的贡献，并探讨了ＫＤＰ作为非线性晶体材料三阶非线性系数测量基准的可行性。     

利用正则化方法求解浅海声波散射问题

讨论了浅海中波导的声波散射问题，我们将该问题归结为一个第一类边界积分方程，利用正同化方法求解，并证明了该方法的收敛性，数值例子表明了该方法的简单与有效性。     

The multiple-scale wave equation

An analytic and numerical study of the behavior of the linear nonhomogeneous wave equation of the form ε²u_tt = Δu + tf with high wave speed (ε 1) is carried out. This study was initially motivated by meteorological observations which have indicated the presence of large spatial scale gravity waves in the neighborhood of a number of summer and winter storms, mainly from visible images of ripples in clouds in satellite photos. There is a question as to whether the presence of these waves is caused by the nearby storms. Since the linear wave equation is an approximation to the full system describing pressure waves in the atmosphere, yet is considerably more tractable, we have chosen to analyze the behavior of the linear nonhomogeneous wave equation with high wave speed. The analysis is shown to be valid in one, two, and three space dimensions. Partly because of the high wave speed, the solution is known to consist of behavior which changes on two different time scales, one rapid and one slow. Additionally, because of the presence of the nonhomogeneous forcing term tf, we show that there is a component of the solution which will vary only on a very large spatial scale. Since even the linearized wave equation can give rise to persistent large spatial scale waves under the right conditions, the implication is that certain storms could be responsible for the generation of large-scale waves. Numerical simulations in one and two dimensions confirm analytic results.     

Linear and Nonlinear Optical Properties Of Racemic (±)2-(α-Methylbenzylamino)-5-Nitropyridine Single Crystals

Linear and nonlinear optical properties of racemic (±)2-(α-methylbenzylamino)-5-nitropyridine ((±)MBANP) single crystals have been comprehensively investigated and compared with those of the enantiomorph (–)2-(α-methylbenzylamino)-5-nitropyridine ((–)MBANP) crystals. (±)MBANP crystal exhibits very high chemical and physical stability, but relatively small nonlinear optical coefficients (d₃₁ = 6.8 pm/V, d₃₂ = 4.7 pm/V, d₃₃ = 0.84 pm/V). A comparison between the nonlinear optical coefficients of (±)MBANP and (–)MBANP demonstrates the validity of the oriented-gas model in molecular crystals that neglects all the contributions from intermolecular interaction.     

一类非线性单调型方程的区域分裂法

本文考虑了一类非线性单调问题的加性Schwgrz交替法和异步平行算法,并得到了在能量模意义下的收敛性结果,最后还讨论了格式的有限元离散。     

谐振子薛定谔方程的简单解法

物质的许多物理与化学性质都可以用线性谐振子模型解释，本文用简单的数学运算求解线性谐振子的薛定谔方程，避免了特殊函数等复杂的数学运算，得出了量子力学教材完全相同的结果。