非线性偏微分方程的非线性微分约束

项目名称： 非线性偏微分方程的非线性微分约束

项目编号： No.11301007

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 朱春蓉

作者单位： 安徽师范大学

项目金额： 22万元

中文摘要： 不变子空间方法、分离变量方法和广义条件对称方法都可以用微分约束方法进行解释。通过该方法可以约化大量非线性偏微分方程或构造它们的解。本项目将在不变子空间方法的研究基础上，构建非线性偏微分方程的非线微分约束：(1)偏微分方程允许的不变子空间可以看作是由其线性微分约束定义的,在此基础上提出偏微分方程的各种非线性微分约束；(2)研究非线性偏微分方程的非线性微分约束问题中的理论问题；(3)研究与一些具体的非线性偏微分方程相容的非线性微分约束，从而得到它们的解或将它们约化为有限维动力系统，并研究解的相关性质；(4)结合分离变量方法,研究与高维偏微分方程相容的非线性微分约束及其相关问题。

中文关键词： 非线性偏微分方程；非线性微分约束；不变子空间方法；李对称方法；

英文摘要： Many symmetry methods and related methods can be reformulated by using the technicalities of differential constraints method. By the method of differential constraints, exact solutions of some nonlinear partial differential equations (PDEs) were constructed, and some nonlinear PDEs were reduced to finite-dimensional dynamics. On the basis of the research of invariant subspaces method, this project is aimed at constructing nonlinear differential constraints of nonlinear PDEs. Firstly, definitions of various classes of nonlinear diffential constraints of PDEs will be presented. As is well known, invariant subspaces admitted by nonlinear PDEs are defined by their compatible linear differential constraints. Secondly, theory of nonlinear differential constraints will be studied. Thirdly, nonlinear differential constraints compatible with some nonlinear PDEs will be given. As a consequence, solutions of corresponding PDEs are constructed, or the corresponding PDEs are reduced to finite-dimensional dynamics. Then behaviors of the resulting solutions will be discussed. Finally, combining the method of separation of variables and differential constraints, nonlinear differential constraints compatible with multi-dimensional partial differential equations and related problems will be studied.

英文关键词： nonlinear differential partial equations；ninlinear differential constranints；invariant subspace method；Lie symmetry method；