李超代数上符号计算问题及其在超可积系统中应用

基于有限维Lie超代数和对应loop代数，利用屠格式构造新矩阵谱问题以及相联系的超非线性演化方程族机械化算法。我们将把挑选超对称孤子方程推广到超对称理论上去。超演化方程族的代数几何性质，包括具有超Hamilton结构、对称约束下的有限维可积系统和无穷多守恒律将通过计算机代数的方法给与分析。机械化的算法与多样的软件是研究矩阵谱问题和超演化方程Liouville可积性的重要工具。与此同时超非线性演化方程（组）对称问题也是数学、力学及工程中经常遇到的重要问题。在研究新方程对称过程中，会出现新的、大量的超定偏微分方程组，应用吴微分特征列和符号计算是研究此问题重要工具。编制程序并找到新的对称是解决对称问题的关键，同时也为对称问题奠定理论与技术基础。因此通过本项目研究，不仅使非线性超可积系统算法和理论得到提升，强化多学科间相互交叉，促进微分形式吴方法进一步发展。

Based on finite-dimensional Lie super algebras and their corresponding loop algebras, we will construct new matrix spectral problems and their associated hierarchies of super nonlinear soliton equations by using the mechanization method and the generalized Tu scheme. We will also single out supersysmmetric soliton equations to extend the supersymmetric soliton theory. Algebro-geometric properties of super soliton equations, including super Hamiltonian structures, symmetry constraints and conservation laws, will be analyzed by using computer algebraic proofs. The mechnization algorithm and diverse software packages are the fundamental tool to study the appropriateness of matrix spectral problems and Liouville integrability of super nonlinear soliton equations generated from the compatibilty conditions of the matrix spectral problems. Moreover, we will adopt the Wu method and symbolic computations to deal with overdetermined parial differential equations, appearing in the study of the symmetry problem of super soliton equations, which is often encountered in mathematics, mechanics and engineering sciences. All theories by computer algorithms and programs in treating the symmetry problem will also form the theoretical and technical foundation for our research. To conclude, through this project, not only new algorithms and theories of super nonlinear integrable equations, inlcuding generalizization of the Wu method of differential form, will be developed, but also interdisciplinary research and collaborative learning will be promoted.