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A new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling. (English) Zbl 1063.37068
Summary: A type of new interesting loop algebra $$\widetilde G_M$$, $$M = 1,2,\dots$$, with a simple commutation operation just like that in the loop algebra $$\widetilde A_1$$ is constructed. With the help of the loop algebra $$\widetilde G_M$$, a new multicomponent integrable system, M-AKNS-KN hierarchy, is worked out. As reduction cases, the M-AKNS hierarchy and M-KN hierarchy are engendered, respectively. In addition, the system 1-AKNS-KN, which is a reduced case of the M-AKNS-KN hierarchy above, is a unified expressing integrable model of the AKNS hierarchy and the KN hierarchy. Obviously, the M-AKNS-KN hierarchy is again a united expressing integrable model of the multicomponent AKNS hierarchy (M-AKNS) and the multicomponent KN hierarchy(M-KN). This article provides a simple method for obtaining multicomponent integrable hierarchies of soliton equations. Finally, we work out an integrable coupling of the M-AKNS-KN hierarchy.

##### MSC:
 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
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