【作者】 张卓；

【导师】 唐翌；

【作者基本信息】 湘潭大学 ， 理论物理， 2002， 硕士

【摘要】 孤子理论是非线性科学的一个重要分支。在数值模拟和实验中，人们直观的观察到了孤子象准粒子一样，相互穿透后能保持其物理性质不变。后来，发展了一种巧妙而复杂的数学方法，称为逆散射方法，用来普遍求解可积系统的孤子方程。 从上面公认的物理图象和数学结果出发，我们从两个方面研究了孤子的相互作用问题。这一课题，显而易见，在孤子的动力学理论方面具有基本意义。而且，在应用中，光孤子的相互作用是对光纤孤子通信的一个根本性的限制。另外，新近发现，在一个典型的非线性系统里，孤子和杂质的共同作用，导致了一个抑制混沌的迷人前景。在本文中，我们首先以已被证实的孤子图象为根据，比较详细的研究孤子之间的相互作用。明确阐明了孤子碰撞过程的物理机制。结果表明KdV方程的孤子有较弱的相互吸引。随后，我们仔细考察了一维参数激发的非线性振子阵列，这是一个典型的非线性物理体系，其中孤子和杂质的共同作用，能在体系的混沌参数区域导致时空有序的图案。我们用多重尺度方法，推导出了它的振幅方程。全文安排如下：在引言部分，我们描述了非线性科学研究的概貌，介绍了研究背景。其中包括混沌和孤子的研究历史，直观概念以及它们的重大应用，从中说明我们选题的的动机。第一章是基本理论，详细介绍了孤子的公认的重大理论结果，我们研究工作的数学工具和相关的实验技巧。第二章详细讨论了孤子之间的相互作用的物理机制。第三章研究了孤子和杂质的相互作用。在最后一章是总结和展望。更多还原

【Abstract】 Numerical simulations and experiments, it was revealed that solitons can mutually punctuate through each other like mass point, remaining their own features. Later, a very sophisticated method, namely, the method of inverse scattering transformation, was developed to commonly solve nonlinear equations possessing the solution of solitons.From the physical picture and mathematical results mentioned above, we study the problem of solitons’ interaction which is not only valuable for dynamics of solitons, but also set a basic limit for soliton communication in optical fibers. Moreover, in the most recent years, a fantastic prospect of controlling chaos was demonstrated in a typical nonlinear system due to the cooperation of solitons and disorder. In the present paper, grounding on the convincing pictures of soliton, we begin with studying the interaction between solitons, definitely setting forth the physical mechanism of collision between solitons. Secondly, we thoroughly investigate an array of parametrically driven nonlinear oscillators, in which spatiotemporal patterns are formed despite the chaotic parameters when both solitons and impurities are present. By the virtue of method of multiple scales, we have derived the amplitude equation. The paper has been organized as following: in the part of introduction, we roughly describe the general picture the nonlinear science. This part consists of history of studying chaos and soliton, intuitive illustration about their concept and important application. In the first chapter, the fundamental results on the theory of solitons, along with related techniques in mathematics and experiment are introduced. The following two chapters deal with the problem of interaction<WP=16>between solitons and between solitons and impurities, respectively. We spend the last chapter on summary and prospect about theory of solitons.更多还原