非线性演化方程的对称约化和不变子空间

The nonlinear phenomena widely appear in almost all the scientific fields such as mathematics, chemistry, biology, communication, society, economy, and so on. As a result, to deal with the nonlinear equations that describe the nonlinear systems becomes one of the main and hot topic for the researchers. It’s well known that symmetry reduction method related to group theory is an effective tool to the study of exact solutions for the nonlinear partial differential equations. The classification and symmetry reductions for nonlinear evolution equations are discussed by using the method of combing conditional Lie-Backlund symmetry with invariant subspaces theory. At the same time, exact solutions for some typical models are obtained. Through studying and analysising the behavior of the solutions, we portray inner nature of the equations. As a result, we can explain essential characteristics of nonlinear phenomena.The following nonlinear evolution equations will be considered in the project. (1) Studying Hamilton-Jacobi equations , which comes from the field such as ,optimal control theory, differential geometry, classical mechanics, quantum mechanics, and so on. (2) Nonlinear diffusion equations with x-dependent diffusion and source terms and the famous Black-Scholes Option Pricing Model are discussed.(3) KdV type of nonlinear evolution equations are investigated. Researches in the project have a strong practical application background.

非线性现象广泛地呈现在物理、化学、生物、通讯、社会、经济等领域。对于描述非线性现象的非线性偏微分方程的精确解以及探讨方程解所具有的特性的研究已经成为众多学者研究的热点。与对称群相关的方法是研究非线性偏微分方程精确解的有效方法。本项目运用条件Lie-Backlund对称与不变子空间理论相结合的方法对一类非线性演化方程进行对称约化，同时对方程进行分类，求得典型模型的精确解。并通过对方程解的性态的研究和分析，刻画方程的内在性质，解释非线性现象的本质特征。本项目拟对如下非线性演化方程进行研究。（1）研究来源于最优控制理论、微分几何、古典力学、量子力学等领域中的Hamilton-Jacobi方程。（2）研究反应项和热源项依赖于x的二阶非线性扩散方程，以及著名的Black-Scholes期权定价模型。（3）研究KdV型非线性演化方程。本项目所研究的问题具有很强的实际应用背景。

本项目基本按照原计划执行。运用条件Lie-Backlund对称和不变子空间理论相结合的方法，研究了金融数学中重要的模型之一Black-Scholes方程的对称约化，讨论了变系数的反应扩散方程的对称约化和方程的分类问题，对于三阶非线性演化方程的对称约化问题做了初步的探讨和研究。