Two New Integrable Hierarchies and Their Nonlinear Integrable Couplings

By introducing an invertible linear transform, a new Lie algebra G is obtained from the Lie algebra H. Making use of the compatibility conditions of the respective isospectral problems, a generalized NLS-MKdV hierarchy and a new integrable soliton hierarchy are achieved by using the trace identity or the variational identity. Then, two special non-semisimple Lie algebras  and  are explicitly conducted. As an application, the nonlinear continuous integrable couplings of the obtained integrable systems as well as their bi-Hamiltonian structures are established, respectively.