从FPU问题到孤立子

From FPU Problem to Solitons

摘要: 孤立子，混沌和分形是非线性科学的重要组成部分，在上世纪得到了极大的发展。在非线性科学的创立和发展过程中，Fermi-Pasta-Ulam (FPU)问题扮演着及其关键的作用。本文首先简要介绍FPU问题，分析与其相关的线性模型的解法和简正模式的能量。然后从FPU问题导出Korteweg-de Vries (KdV)方程，求出单孤立子解，给出了单孤立子和双孤立子的图形。

Abstract: Solitons, chaos and fractals were important parts of nonlinear science, which had been invented and seen a great many developments in the last century. During the developments of the nonlinear science, Fermi-Pasta-Ulam (FPU) problem played a crucial role. We will introduce the FPU problem briefly here and analyze the linear mathematical model and the energy of normal modes that are related to FPU problem. At the same time, we will explain the process of deriving the KdV equation from FPU problem and get the one-soliton solutions to the KdV equation. Also the plots for one-soliton and two-soliton solutions are presented.