非线性波方程的可积离散、非局域对称和保可积数值算法

项目名称： 非线性波方程的可积离散、非局域对称和保可积数值算法

项目编号： No.11275072

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 陈勇

作者单位： 华东师范大学

项目金额： 80万元

中文摘要： 离散与连续是现实世界物质运动对立统一的两个方面.离散模型和连续模型是描述、刻画和表达现实世界物质运动的两种有力工具，它们既相互通达，但又有各自的特点.本课题主要研究一类数学物理中非常重要的波动方程:流体力学中描述浅水表面波的CH 方程和DP方程；液晶中定向波的HS方程；海洋中考虑到地球旋转效应的简化Ostrovsky方程及超短脉冲方程等.其共同特点:可积，但是数学结构很复杂，求解很困难，解存在波爆破现象，需要精度很高的特殊算法.我们通过双线性方法和对称方法对这类方程开展研究:(1)结合双线性和Hodograph变换进行可积离散,构造此类方程的自调整移动网格算法.(2)通过有限变换和对称方法进行对称离散,提出保对称性算法.这将产生一类新的具有奇异解的可积系统，发现奇异性波解,椭圆周期波和孤立波相互作用,为离散可积系统的研究提供新的思路和工具,为模拟具有奇异性孤子间相互作用提供新的数值格式.

中文关键词： 离散可积系统；保对称算法；怪波；对称优化；RHP问题

英文摘要： Discreteness and continuation are two aspects with unity of opposites for the movement of our universe. The discrete models and continuous models are two powerful tools to describe and reflect natural phenomena. They are interrelated to each other whereas each has its own characteristics. In this research project, we are concerned with a class of important nonlinear wave equations in mathematical physics: the CH and DP equations which describe shallow water waves in hydrodynamics; the HS equation of high-frequency waves in liquid crystal; the reduced Ostrovsky equation for ocean waves with rotational effect of the earth and the short pulse equation in nonlinear optics. These equations share some common characteristics: they have complicated mathematical structures though they are integrable; it is difficult to find their exact solutions and there exists blow-up solution. Therefore, special numerical methods with high precision are needed. This project focuses on the study of this class of equations by virtue of bilinear and symmetry methods. To be more specific, we plan (1) to construct integrable discrete analogues and self-adaptive moving mesh algorithm for this class of equations by combing the bilinear forms and hodograph transformation; (2) to provide symmetric discretization through finite transformatio

英文关键词： discrete integrable system；symmetry-preserving algorit；rogue wave；symmetry optimazation；RHP