The multi-component generalized Wadati-Konono-Ichikawa (WKI) hierarchy and its multi-component integrable couplings system with two arbitrary functions

A new simple loop algebra G∼M is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component WKI hierarchy is obtained. Furthermore, an expanding loop algebra F∼M of the loop algebra G∼M is presented. Based on F∼M, the multi-component integrable couplings system with two arbitrary functions of the multi-component WKI hierarchy is worked out. How to design the isospectral problem of the multi-component hierarchy of equations is a technique and interesting topic. The method can be applied to other nonlinear evolution equations hierarchy.