【作者】 戴朝卿；

【导师】 张解放；

【作者基本信息】 浙江师范大学 ， 理论物理， 2005， 硕士

【摘要】 本文主要从单模光纤传输模型—常系数和变系数非线性薛定谔方程出发，运用最近得到发展的求精确解的扩展的双曲函数法、F-展开法、基于AKNS技术的Darboux变换方法等方法和求数值解的Adomian分解方法，求得了孤波解，研究了基本光孤波和双孤波在单模光纤中的传输情况，为进一步实现超高速、大容量的光信息传输提供一定的理论依据。此外还运用扩展的雅可比椭圆函数展开法讨论了可积的微分-差分非线性薛定谔方程（AL模型）。主要内容如下： 第一章主要介绍了光孤子的研究进展。第二章简要地介绍了一些与光孤子有关的基本概念，其中包括光纤结构和传输模、时间孤子和空间孤子的区别、包络孤子和亮、暗孤子、色散效应、非线性效应、脉冲自变陡和孤子自频移现象、以及Gordon-Haus效应等概念。第三章中，对最近得到发展的求精确解的扩展的双曲函数法、雅可比椭圆函数展开法、F-展开法、基于AKNS系统的Darboux变换方法等方法和求数值解的Adomian分解方法进行了简单的介绍。 第四章中首先我们用Adomian分解方法求解了标准非线性薛定谔方程iu_z±1／2u_tt+|u|²u=0，讨论了单孤子及其双孤子的演化行为。结果表明亮、暗单孤子能无畸变地稳定传输；选择合适的初始间距，双孤子能在光纤中彼此互不影响地独立传播。亮、暗单孤子的近似解与解析解间的绝对误差极小，近似解很好地逼近了解析解。这说明Adomian分解方法虽然没有离散化方程而是运用逆算符来求得近似解，但它是一种较好的数值算法。其次用扩展的双曲函数法求解了高阶非线性薛定谔方程u_z=iα₁u_tt+iα₂|u|²u+α₃u_ttt+α₄（|u|²u）_t+α₅u（|u|²）_t，得到了亮、暗孤波解。在一定的参数条件下，我们还运用实振幅求得了该方程的三种组合孤波（W形、M形和亮暗组合孤波），其中W形和M形孤波与文更多还原

【Abstract】 Starting from nonlinear Schrodinger equations with the constant and variable coefficients, we derive the solitary wave solutions of these equations with the help of the methods including the extended hyperbolic function method, the F-expansion method, the Darboux transformation based on Ablowitz-Kaup-Newell-Segur(AKNS) technology and the Adomian decomposition method, and investigate the properties of propagation of fundamental solitary waves and two-solitary waves in monomode optical fiber. What we discussed in this paper may improve the analytic foundation for realizing the transmission of fast-speed and high-capacity optical information systems. Moreover, using the Jacobian elliptic function expansion method, the integrable differential-difference nonlinear Schrodinger equation(AL model) is discussed. The paper is organized as follows:In chapter 1, the research evolution of optical soliton is introduced. Chapter 2 is devoted to presenting some fundamental concepts related to optical soliton, including optical fiber structure and transmission mode, the difference between temporal and spatial solitons, envelop soliton, bright and dark solitons, dispersion effect, nonlinear effect, self-steepening, self-frequency shift, and Gordon-Haus effect, etc. In chapter 3, we briefly introduce some recently developed approaches, such as the extended hyperbolic function method, the F-expansion method, the extended Jacobian elliptic function expansion method, the Darboux transformation based on AKNS technology and the Adomian decomposition method.In chapter 4, we firstly consider the standard nonlinear Schrodinger equation iu_z ± 1/2utt + |u|~2u = 0, and discuss the evolutional behaviors of fundamental soliton and two-soliton solutions. These results indicate that bright and dark fundamental solitons can propagate without distortion. By selecting suitable initial separation between two solitons, they can separately propagate without interaction. The error between approximate solutions and analytical solutions of bright and dark solitons is small, so these approximate solutions approach analytical solutions very well. It demonstrates更多还原