Two hierarchies of new nonlinear evolution equations associated with 3 × 3 matrix spectral problems

Two hierarchies of new nonlinear evolution equations associated with 3 × 3 matrix spectral problems are proposed. The generalized bi-Hamiltonian structures for one of the two hierarchies are derived with the aid of the trace identity. Some explicit solutions of a typical nonlinear evolution equation in the hierarchy are obtained, which include soliton and periodic solutions.