非线性系统的非局域的对称和约束及求解研究

项目名称： 非线性系统的非局域的对称和约束及求解研究

项目编号： No.11271211

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： 李彪

作者单位： 宁波大学

项目金额： 66万元

中文摘要： )：非线性系统的求解研究在非线性科学中极为重要,是国际上热门且前沿的课题。数学机械化的研究为国际自动推理的研究开辟了新的前景，符号和数值计算有机结合已成为研究非线性问题的主要特征。本项目将深入研究非线性系统的非局域对称、非局域约束、精确解和高精度数值解：基于达布变换、贝克隆变换、双线性化等经典方法，探索非线性系统的非局域对称及其局域化方法，揭示非线性科学中一些深层次的内在联系和可能的实际物理应用；深入研究非局域约束理论和方法，给出若干可积系统的新的类型的解或解的结构；把可积系统约化为非线性常微分方程组，通过数值求解这些常微分方程组，获得可积系统的高精度数值解；研究近可积系统，发展一个高精度、可信的数值求解方法。从理论、算法和应用上深入研究上述问题，为实际问题的解决提供新的原理和工具，为数学机械化在非线性领域、如光孤子通信、玻色-爱因斯坦凝聚、流体力学等领域的应用打开新的突破口。

中文关键词： 非线性方程；非局域对称；孤波与椭圆周期波；怪波；符号计算

英文摘要： Constructing solutions for nonlinear systems is very important and a hot research area. The studies of mathematics mechanization open a new window for automated reasoning. The harmonic combination of symbolic and numerical computation is the main character for nonlinear problems. In this project, a thorough study will be carried out on nonlocal symmetries, nonlocal constraints, exact solutions and high precision numerical solutions for nonlinear systems. On the basis of some classical methods, such as Darboux transformation, Backlund transformation, Bilinear method, etc, nonlocal symmetries and their localization methods will be explored for nonlinear systems. Some important relationship and potential real physical applications will be uncovered for nonlinear science. We shall study the theories and methods for nonlinear constraints, give some new types of solutions or solutions' structure for some integrable systems, reduce integrable systems to nonlinear ordinary differential equations (ODEs) and obtain high precision numerical solutions of integrable systems by solving these ODEs. We shall propose a trustable and high precision numerical method for approximately integrable systems. By studying the theories, algorithms and applications for the above problems, some new principals and tools will be provided fo

英文关键词： nonlinear equations；nonlocal symmetries；soliton-Cnoidal wave；Rogue waves；symbolic computation