【作者】 刘海兰；

【导师】 文双春；

【作者基本信息】 湖南大学 ， 通信与信息系统， 2007， 硕士

【摘要】 对于那些在许多非线性问题中起核心作用的孤子方程，寻找它的解无疑是一项非常重要的工作。除了可以用数值方法对某些孤子方程的近似解进行研究以外，寻找孤子方程的精确解对于理解孤子方程的性质有着非常大的帮助。本文基于扩展的双曲级数方法，找到了两个很重要的非线性系统，即非线性管理光纤光栅和超常介质中的精确孤子解，取得的主要成果如下：首先，用扩展双曲级数方法得出了最近提出的新的非线性管理光纤光栅中的精确孤子解。基于考虑了非线性管理效应后的非线性耦合模方程，利用多重展开方法将它简化为微扰的非线性薛定谔方程，再利用扩展双曲级数方法求解这个微扰的非线性薛定谔方程，得到了新的暗孤子解，并分析了影响暗孤子的形成和传播的重要参数，得到了在不同条件下光纤光栅中能产生孤子的最低功率值。其次，利用同样的方法求出了超常介质中超短电磁脉冲传输方程的精确孤子解。超常介质与常规光学介质不同，它具有色散磁导率，这个色散磁导率导致在超短脉冲非线性传输方程中出现可控的自陡效应项和各阶非线性色散项。我们基于得到的精确孤子解，分析了可控自陡效应和二阶非线性色散效应对孤子形成和传输特性的影响。结果表明，超常介质中负的自陡效应使得孤子脉冲的中心位置随传输距离向左漂移，与常规介质中自陡效应(恒为正)的作用相反；特别是，由于二阶非线性色散的作用，在没有线性色散的情形下同样可形成孤子，而且在反常线性色散情形下也可形成暗孤子，这为孤子理论提供了新的视角。更多还原

【Abstract】 It is undoubtedly a very important problem to find the solutions of the soliton equations which play a kernel role in lots of nonlinear problems. Besides the research on approximate solutions of certain soliton equations using numerical method, finding exact solutions of soliton equations will do great help in understanding the properties of soliton equations. Applying the extended Tanh-function expansion method, exact solutions of two very important nonlinear systems, i.e., nonlinearity management fiber Bragg grating and metamaticals, are gained. The main research results are listed below:Firstly, by using an extended Tanh-Function expansion method, exact solitary solutions are obtained in new nonlinearity management fiber Bragg grating, which is proposed recently. Based on the nonlinear coupled mode equation, which takes nonlinearity management into consideration, the nonlinear coupled mode equation is reduced into the perturbed nonlinear Schr(o|¨)dinger equation through using the multiple scale analysis. Then dark solitary solutions can be constructed by an extended Tanh-Function expansion method. Furthermore, the effects of the physical parameters of nonlinear periodic structure on soliton propagation are discussed, and the required minimum power of soliton is given under various conditions in the fiber Bragg grating structure.Secondly, exact solitary solutions of propagation equation of ultrashort pulse are derived in metamaticals (MMs) by employing the same method. Different from ordinary materials, the MM has a dispersive magnetic permeability, which results in a controllable SS effect and a series higher order nonlinear dispersion items in the ultrashort pulse propagation equation. Based on these exact solutions, the influence of the controllable SS effect and second-order nonlinear dispersion on formation and propagation of dark electromagnetic solitons is discussed. It is found that the negative SS effect in MMs makes the soliton center move to the leading side of soliton, opposite to that in ordinary material in which the SS effect is always positive. Most importantly, due to the role of the second-order nonlinear dispersion, dark solitons can be formed in the absence of linear dispersion, or even in the case of anomalous linear group-velocity dispersion, which gives new visual angle to soliton theory.更多还原