TWO-POINT BOUNDARY VALUE PROBLEM FOR A CLASS OF SECOND-ORDER DIFFERENCE EQUATIONS

In this paper, we apply a critical point theorem and obtain the condition for the existence of three solutions to two-point boundary value problem of a second order nonlinear difference equation.     

完全非线性偏微分方程解的Gevrey微局部正则性

本文中，我们首先简要回顾了Gevrey类中的仿微分运算，然后考察了相关的完全非线性偏微分方程的象征的一些性质。作为应用，我们得到解在椭圆点附近的Gevrey微局部正则性。     

高斯型空间光孤子相互作用的数值研究

 通过数值模拟的方法，对高斯孤子在对数型饱和非线性介质中的相互作用进行了研究，考查了两光束间的相对振幅和相对相位对其相互作用的影响。结果表明：高斯孤子之间的相互作用敏感地依赖于两光束间的相对振幅和相对相位。在不同的振幅差异范围内，光束间的主要作用交替地表现为相互排斥和相互吸引，并由于高斯孤子的不稳定性，导致了光束在碰撞后以一种尺寸周期性变化的呼吸模式传输。随着相对相位的增大，两光束间始终持续地表现出强烈的排斥作用，直到相对相位增加到一个2π周期之后。而且碰撞之后，光束也都以呼吸模式进行传输，其分离的角度越大，呼吸就越明显。     

飞秒孤子传输演化的数值研究

利用分步傅里叶法数值模拟了飞秒孤子在光纤中的传输演化过程。对光纤中单个孤子的传输及二阶孤子的自陡峭效应和自频移效应进行了分析,指出在一定的参数取值范围内,自频移效应对二阶fs孤子传输的影响要比自陡峭效应大,占主导地位,且对自陡峭效应有一定的抑制作用。     

级联三能级介质中非线性光学信号的增强

揭示非最大相干可能比最大相干产生更有效的非线性频率转换. 为此以三能级级联系统(没有最大相干)为例计算其非线性光学信号的强度, 并与三能级Λ型系统(具有最大相干)进行比较. 利用非微扰方法计算表明, 在双光子共振和相同参数的条件下, 级联系统产生的非线性光学信号的最大强度约是Λ系统的两倍. 利用缀饰态表象分析表明, 两个系统中耦合跃迁的自发辐射及其介入的不可逆三光子过程具有相反的作用. 它们在级联系统中加强非线性转换, 而在Λ系统中削弱非线性转换. 关键词： 电磁感应透明 非线性光学过程 双光子共振     

Nonlinear Effects for Bose Einstein Condensates in Optical Lattices

Cold atoms and, more recently, Bose-Einstein condensates （BEC＇s） in optical lattices have attracted increasing interest since their first realization. In particular, the formal similarity between the wavefunction of a BEC inside the periodic potential of an optical lattice and of the electrons in a crystal lattice has triggered theoretical and experimental efforts alike. Many phenomena from condensed matter physics, such as Bloch oscillations and Landau-Zener tunneling have been shown to be observable also in optical lattices. An important difference between electrons in a crystal lattice and a BEC inside the periodic potential of an optical lattice is the strength of the self interaction and hence the magnitude of the nonlinearity of the system. Electrons in a metal are almost noninteracting, whereas atoms inside a BEC interact strongly. A＇ perturbation approach is appropriate in the former case while in the latter the full nonlinearity must be taken into account. From this feature new physics is expected. Most experiments to date have been carried out in the regime of shallow lattice depth, for which the system is well described by the mean field Gross-Pitaevskii equation with a periodic potential. Moreover, the nonlinearity induced by the mean-field of the condensate has been shown, both theoretically and experimentally, to give rise to instabilities in certain regions of the Brillouin zone. These instabilities are not present in the corresponding linear system, i.e. the electron system. Experimental and theoretical results on the subject of nonlinear Landau-Zener tunneling and nonlinearity-induced instabilities in a Bose-Einstein condensate interacting with an external periodic potential will be presented.     

等幂代数和方程组的解法

众所周知, 等幂和代数方程组可以通过Newton恒等式转化为一个高次代数方程. 这就是Viete-Newton定理. 本文报道一项关于把Viete-Newton定理推广到等幂代数和方程组的求解上去的研究成果. 利用代数学和组合学的知识和技巧, 该成果显示了等幂代数和方程组可以封闭地转化为两个次数之和等于等幂代数和方程组未知数个数的代数方程.     

从诺贝尔物理学奖看光学的发展

从一个新的角度分析、讨论了光学的发展过程.并对光学今后的发展做了展望.     

解奇异非光滑方程组的牛顿法和不精确牛顿法

本文主要解决奇异非光滑方程组的解法。应用一种新的次微分的外逆，我们提出了牛顿法和不精确牛顿法，它们的收敛性同时也得到了证明。这种方法能更容易在一引起实际应用中实现。这种方法可以看作是已存在的解非光滑方程组的方法的延伸。