An isospectral transformation and the related integrable hierarchies of soliton equations

By making use of an isospectral transformation, an isospectral matrix with non-zero trace is turned into a zero-trace one. With the help of the simple loop algebra A1 and the Tu scheme, a soliton-equation hierarchy is generated under the framework of the zero curvature equation. By employing the trace identity, its Hamiltonian structure is obtained. Then, we enlarge the Lie algebra A1 into three kinds of Lie algebras, which are devote to investigating integrable couplings. The corresponding three types of integrable couplings are worked out, one of them has Hamiltonian structure which is obtained by the quadratic-form identify. The approach for generating integrable couplings has extensive applications.