A New Lie Algebra and Its Related Liouville Integrable Hierarchies

A new Lie algebra G and its two types of loop algebras G1 and G2 are constructed. Basing on G1 and G2, two different isospectral problems are designed, furthermore, two Liouville integrable soliton hierarchies are obtained respectively under the framework of zero curvature equation, which is derived from the compatibility of the isospectral problems expressed by Hirota operators. At the same time, we obtain the Hamiltonian structure of the first hierarchy and the bi-Hamiltonian structure of the second one with the help of the quadratic-form identity.     

Two Types of Expanding Lie Algebra and New Expanding Integrable Systems

From a new Lie algebra proposed by Zhang, two expanding Lie algebras and its corresponding loop algebras are obtained. Two expanding integrable systems are produced with the help of the generalized zero curvature equation. One of them has complex Hamiltion structure with the help of generalized Tu formula （GTM）.     

广义KN方程族及其Hamilton结构

构造了一个带有任意光滑函数的等谱问题,利用屠规彰格式得到广义KN 方程族及其Hamilton结构,并且当f=0时,变为著名的KN谱,当f=-12qr时,变为Qiao谱.