一个新的Lie代数和它的应用

摘要

通过构造一个新的Lie 代数，利用它相应的Loop 代数设计等谱Lax 对，根据其相容性条件，得到了一族 Lax 可积方程族，其一种约化形式为著名的 AKNS 族。根据迹恒等式得到该方程族的Hamilton 结构。利用该可积方程族可以进一步研究它的达布变换、对称、代数几何解等相关性质。%By constructing a new Lie algebra and its corresponding Loop algebra, an isospectral Lax pair is established whose compatibility condition gives rise to a Lax integrable hierarcy, whose reduced form is the well-known AKNS hierarchy. Its Hamilton structure is obtained by the use of the trace identity. Then, its Darboux transformations, symmetry, algebro-geometric solutions, and so on will be investigated further.