Department of Mathematics,

Bohai University,

Jinzhou, Liaoning Province,

121000 P.R. CHINA

E-mail: xiatc@yahoo.com.cn, xiatc@student.dlut.edu.cn

**The multi-component generalized Kaup-Newell hierarchy
and its integrable couplings system with two arbitrary functions**

**Abstract:**

We devise a new simple loop algebra $\tilde{G}_M$ and an isospectral
problem. By making use of Tu scheme, the multi-component generalized Kaup-Newell

hierarchy is obtained. Furthermore, an expanding loop algebra $\tilde{F}_M$
of the loop algebra $\tilde{G}_M$ is presented. Based on $\tilde{F}_M$,
the multi-component integrable couplings system with two arbitrary functions
of the multi-component generalized Kaup-Newell hierarchy are worked out.
The method can be applied to other nonlinear evolution equations hierarchy.