Multi-component Levi Hierarchy and Its Multi-component Integrable Coupling System

A simple 3M-dimensional loop algebra X is produced, whose communitation operation defined by us isas simple and straightforward as that in the loop algebra A;. It follows that a general scheme for generating multi-component integrable hierarchy is proposed. By taking advantage of X, a new isospectral problem is established, andthen by making use of the Tu scheme the well-known multi-component Levi hierarchy is obtained. Finally an expandingloop algebra F;of the loop algebra X is presented, based on the F;, the multi-component integrable coupling system ofthe multi-component Levi hierarchy is worked out. The method in this paper can be applied to other nonlinear evolutionequation hierarchies.