A Super Integrable Hierarchy and Its Nonlinear Super Integrable Hamiltonian Couplings*

Basing on an existing Lie super algebra, we introduce a kind of a new Lie super algebra G and establish two isospectral Lax matrices. By taking use of the Tu scheme and the super trace identity, a super integrable hierarchy along with Hamiltonian structure is deduced, from which some evolution equations are obtained, including the well-known mKdV equation. Then, we employ the Lie super algebra G to extend a new higher-dimensional Lie super algebra G¯, from which the nonlinear super integrable couplings of the super integrable hierarchy are given. Moreover, the super trace identity over the corresponding loop super algebra is used to furnish the super Hamiltonian structure for the resulting nonlinear super integrable couplings.