非线性波方程的可积离散、非局域对称和保可积数值算法

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 transformation and symmetry and to propose the symmetry-preserving algorithm. Through the completion of the project, we aim at producing a new class of integrable systems with singular solutions, finding these singular wave solutions, and exploring the interaction of elliptic periodic waves with solitary waves. Moreover, it can provide us with new thoughts and tools for studying discrete integrable systems, and new innovative numerical schemes to simulate the interactions among solitons with singularity.

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.

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。怪波现象研究由于其重要实用价值是目前波动研究的热点之一。对称优化一直是我们研究的重点和突破口。而RHP问题由于其深刻的数学内涵和强有力的工具，是目前研究的挑战性课题。非线性系统程序包的开发是我们研究团队的特色。 本课题通过双线性方法和对称方法对这类方程开展研究:利Reciprocal变换进行可积离散,构造此类方程的自调整移动网格算法.通过有限变换和非线性化方法进行对称离散,提出保对称性算法。我们按照计划在这两个方面完成了设定任务。并在以下几个方面开展了研究工作。1.可积系统的可积离散和自适应数值算法；2. 非线性系统保对称离散算法研究；3.怪波的数学理论及其应用；4. KP方程族约化在可积系统孤子解、怪波解中的应用；5. 活动标架算法构造微分不变量; 6.对称优化的研究; 7.软件包的开发；8.RHP问题的数值求解研究。我们研究了大量的重要数学物理方程，发现了孤立波，lump波，椭圆波，共振波，各种怪波，及其他们相互作用。这对认识一些重要的物理现象，如对海浪和非线性光学的基本了解具有积极的影响。并对应用数学、计算数学、物理学和工程学有所贡献。我们开发了5个基于maple程序下的软件包，可以作为非线性科学的研究工具。在国际重要学术期刊共发表SCI论文51篇。培养博士生10名。获得国家公派基金出国两名。其中5人博士生和2名硕士生获得国家级奖学金。两名上海优秀博士毕业生。