Toda-like晶格方程基于微分形式方法的对称性研究

In the area of nonlinear science, the study of symmetry analysis of differential equations has been attracting more and more. Lie symmetries and many properties of differential equations are closely related, so lots of scholars extend the classical Lie symmetries and try to find more symmetries. Most calculations for symmetries of differential equations are done with the classical methods, such as the classical Lie group method, the non-classical Lie group method, and the Clarkson and Kruskal's direct method. However, “differential form method” which is based on a differential form technique is a geometrical approach, nowadays, the study of the differential form method is limited in classical Lie symmetries of the differential equations, so we will extend and improve the method to calculate the symmetries of Toda-like Lattices. The plan of the present project is as follows: (1) Extend the method to differential-difference equations with variable coefficients, and try to get more types of symmetries, such as discrete potential symmetries, discrete approximate symmetries, etc.; (2) Give a software package of this method to calculate the symmetries of differential-difference equations. Compared to the differential eauations, finding an effective algorithm to calculate the symmetries of discrete equations is more complicated and difficult. The study of our project shows that the differential form method provides a direct and more powerful mathematical tool for mechanization in calculating the symmetries of nonlinear differential-difference equations.

在非线性科学领域中，关于微分方程对称分析的研究愈来愈受到重视。由于李对称和微分方程的许多性质密切相关，大批学者对经典李对称进行推广，试图找到更多的对称。求微分方程的对称大多数都是通过经典李群法、非经典李群法和 CK 直接法等传统方法得到的。而“微分形式法”是一种基于微分形式的几何方法，目前对于这种方法的研究局限于连续方程的经典李对称，本项目将对这一方法加以推广和完善，用于计算 Toda-like 晶格方程的对称，主要研究：(1) 将微分形式方法推广应用于变系数微分差分方程，并试图获得 Toda-like 晶格方程更多类型的对称，如离散的势对称、近似对称等；(2) 给出软件包，用于机械化计算微分差分方程的对称。相对于微分方程，寻求高效可行的算法来计算离散方程的对称总是更加复杂和困难，本项目的研究可以揭示微分形式方法是机械化计算非线性微分差分方程对称的直接而有效的工具。

在现代数学及其分支、物理和其它科学领域中, 关于微分方程对称分析的研究已经成为关注的焦点, 例如寻找对称、对称变换群﹑对称约化和构造群不变解等等。项目组成员针对几类微分方程进行了系统的研究，主要工作如下：. 1. 研究了(2+1)-维 Caudrey-Dodd-Gibbon-Kotera-Sawada 方程及其 Lax 对的对称约化，分析它们的对称群并利用得到的对称约化原方程和 Lax 对，获得了三组约化方程和它们新的 Lax 对，并给出原方程的群不变解。该成果发表在 Journal of Applied Mathematics (SCI检索)。. 2. 推广微分形式方法应用于变系数微分差分方程。基于离散外微分理论, 利用微分形式方法得到了两个 (1+1)-维 Toda-like 晶格方程的对称, 并将这一方法应用于计算一个(1+1)-维非齐次变系数微分差分方程的对称。. 3. 研究了两个 (2+1)-维 Toda-like 晶格方程的对称变换群，并利用得到的对称变换群分别获得了这两个方程新的类孤子解和类周期解，同时讨论了 Toda 晶格方程新解和旧解之间的关系。