Intermediate Self-similar Solutions of the Nonlinear Schrodinger Equation with an Arbitrary Longitudinal Gain Profile

We study pulse propagation in a normal-dispersion optical fibre amplifier with an arbitrary longitudinal gain profile by self-similarity techniques. We show the functional form of the development of low-amplitude wings on the parabolic pulse, which are associated with the evolution of an arbitrary input pulse to the asymptotic parabolic pulse solution. It is found that for the increasing gain the amplifier output corresponding to the input Gaussian pulse converges to the asymptotic parabolic pulse solution more quickly than the output obtained with the input hyperbolic secant pulse, whereas for the decreasing gain the input pulse profiles have nearly no effect on the speed of convergence to the parabolic pulse solution. These theoretical results are confirmed by numerical simulations.     

A Weakly Nonlinear Model for Kelvin-Helmholtz Instability in Incompressible Fluids

A weakly nonlinear model is proposed for the Kelvin-Helmholtz instability in two-dimensional incompressible fluids by expanding the perturbation velocity potential to third order. The third-order harmonic generation effects of single-mode perturbation are analyzed, as well as the nonlinear correction to the exponential growth of the fundamental modulation. The weakly nonlinear results are supported by numerical simulations. Density and resonance effects exist in the development of mode coupling.     

Precipitate Contribution to the Acoustic Nonlinearity in Nickel-Based Superalloy

The influence of γ′ precipitate on the acoustic nonlinearity is investigated for a nickel-based superalloy, which is subjected to creep deformation. During creep deformation, the cuboidal γ′ precipitate is preferentially coarsened in a direction perpendicular to the applied stress axis. The length and shape factor of the γ′ precipitate increase with creep time. The increase of relative acoustic nonlinearity with increasing fraction of creep life is discussed in relation to the rafting of γ′ precipitate, which is closely related to the scattering and distortion of the acoustic wave     

微分中值问题中辅助函数的构造程式

给出解决微分中值问题时，所需辅助函数的构造程式，并通过实例加以详细解释．所给程式具有一定的可操作性，可帮助学生掌握同类问题解决方案中的规律性．     

用HP图形计算器探索枪械后坐力问题

一、问题导入驳壳枪是整整几代中国人的一项集体记忆．不知你是否注意过，电影中的革命英雄们在开枪杀鬼子锄汉奸时总是横向握枪．这是为什么呢？经过讨论分析，我们觉得这是后坐力的结果，那么后坐力又是怎么一回事呢？通过查阅资料可得，后坐力产生的原因，从根本上讲是动量守恒的结果，即枪炮发射时产生的瞬间后退力量．大家对这个问题非常的感兴趣，并且决定合作搜集一些这方面的资料，了解研究、探讨一下这方面的知识．     

竞赛中的数列问题

数列是初等数学与高等数学的重要衔接点，它既具有函数特征，又能构成独特的递推关系，使得它既与高中数学其它部分的知识有着密切的联系，又有自己鲜明的特征，一直是高考考查的重点和热点，也是各类竞赛考查的重点，是考查函数与方程、转化与化归、分类讨论、配方法、换元法、待定系数法等基本数学思想方法的理想载体．     

问题征解

编者按：本栏目精选适合高中学生的有趣、实用、新颖、灵巧、深浅适度、富有启发性的题目进行征解，使其成为启迪思维、开发智力的小智囊．该栏目面向广大读者征集问题，问题的选题范围不做限制，但难度应适当控制，适宜高中学生解答,     

长方体同心球的几条性质

《数学通报》数学问题1813是: 正方体内切球的半径为R,P为球上任一点,P到正方体各面的距离分别为PNi(i=1,2,…,6).证明:∑6i=1PN2i=8R2,∑6i=1PN3i=12R3. 此题凸显了长方体同心球的一些性质. 所谓长方体的同心球,是指球心在长方体中心的球,长方体的外接球是它的特例,当长方体正好是正方体时,其内切球也是它的同心球.