非等谱可积方程族的对称、群表示理论以及尖孤子解

首先本项目提出了生成非等谱可积方程族的两种方法:第一种是Lax矩阵法，通过选取适当Lax矩阵和谱演化形式，利用屠格式和自对偶的Yang-Mills方程及其约化方程构造新的可积方程族；第二种是构造新的非等谱拟微分算子李代数，利用R-矩阵法构造李代数g的对偶李代数g*上适当的Lie-Poisson括号，并利用广义Lax方程寻求可积方程族。其次本项目利用所得到的Lie-Poisson括号的Casimir泛函，探求得到的可积方程族的Hamilton结构，通过约化方程族得到具有物理意义的单偏微分方程，利用群表示理论研究其新对称，并寻求非等谱方程的尖孤子解。最后本项目利用谱形变方法研究得到的非等谱方程族的K对称、τ对称及其李代数结构。至此本项目不仅提出了具有实际物理意义的非等谱可积方程族，而且研究了所得可积方程族的对称性质、对称的群表示及求解方法，为理论物理学的研究提供了一整套有效的数学工具。

The methods for generating the non-isospectral integrable hierarchies in the item. One way is the Lax-matrix method, which employs the formats of spectral evolution properly operators chosen and the Tu scheme and the self-dual Yang-Mills equation and its reduced equations to generate new integrable hierarchies. Another way is to employ R-matrix method to construct some new pseudo-differential operators with the nonisospectrals, which refers to the Lie algebra method. By using the constructed Lie algebra g and its dual Lie algebra g*, we choose the appropriate Lie-Poisson brackets on g* to deduce some new nonisospectral integrable hierarchies by the generalized Lax equations. Furthermore, we seek the Hamiltonian structures of the hierarchies obtained via the use of the Casimir functional of the Lie-Poisson brackets. By reduction of the integrable hierarchies, we hope to get some new single PDEs with some physics senses, whose new symmetries are obtained by the theory of group representations. Furthermore, we still seek for new peakon solutions of the PDEs obtained. Finally, we apply the spectral deform method to deduce the K symmetries and the τ symmetries and their Lie-algebra structures of the nonisospectral integrable hierarchies obtained. Thus, we not only propose nonisospectral integrable hierarchies with physical backgrounds, but also investigate the symmetrical properties, the representations of symmetry groups and approaches for solving the integrable equations obtained, which provide efficient mathematical tools for the study of theoretical physics.