李代数分裂框架下的可积族、Backlund变换和几何性质

As one of the mainstream direction of nonlinear science, soliton theory and integrable system has a broad prospects of development and applications in the field of science and technology . It is an important research topic to construct more new nonlinear integrable soliton hierarchies in the soliton theory and integrable systems. Making good use of Lie algebra and loop group as a theoretical tool, through symbolic computation, we try to extend the method of Lie algebra splitting. Based on contructing some new Lie algebra splittings, we are going to construct more new integrable hierarchies. In a unified framework of Lie algebra splitting, we try to obtain the Hamiltonian structures,recursion operators,B?cklund transformations and multi-soliton solutions. Furthermore, we put forward the geometric interpretation together with the invariant curve flows and soliton submanifolds for the hierarchies. This project not only provides a new method of constructing integrable hierarchies,the Lax pairs and Backlund transformations, but also gives the theoretical basis and tools to better check the integrability and to seek the exact solutions of nonlinear differential equations. It puts forward a new idea to research the wonderful algebraic structures and intrinsic geometric properties of the integrable soliton hierarchies, which provides a new tool to explain and solve a lot of practical problems.

孤立子与可积系统作为当今非线性科学研究的主流方向之一，在科学和科技领域有广阔的发展和应用前景。构造更多新的非线性可积系统是孤立子与可积系统理论中一个重要的研究课题。 本项目以李代数和loop群为理论工具,借助于符号计算，扩展李代数分裂方法，通过构造新李代数分裂构造出新的可积方程族，并在李代数分裂的统一框架下构造其B?cklund变换、Hamilton结构、递推算子及多孤子解，且给出可积族的几何解释，得到其不变曲线流和孤子子流形及性质。本项目提供了构造可积方程族及其Lax对和B?cklund变换的新方法，为更好地判定非线性微分方程可积性和求解提供理论依据和方法，为研究孤子可积族的美妙的代数结构和内蕴几何性质提供新思路，为解释和解决实际问题提供新的工具。

构造更多新的非线性可积系统是可积系统领域中一个重要的研究课题。 本项目以李代数和loop群为理论工具,借助于符号计算，扩展李代数分裂方法，通过在标准李代数分裂中加入适当的形变算子的方式构造更多新的李代数分裂，得到了sl(4)-B, so(5)-B 和 sp(2)-B、G2 型扩展李代数分裂及其相应的可积族，并得到了其Bäcklund变换、Bi-Hamilton结构、多孤子解；研究了与这些可积组相关的耦合方程组的求解与解的性质，构造了一类高维微分方程的微分不变量和双线性可积性。本项目提供了构造可积方程族及其Lax对的新方法，为更好地研究非线性微分方程可积性和求解提供理论依据和方法。