Two expanding integrable systems of the GI soliton hierarchy and a generalized GI hierarchy with self-consistent sources as well as its extension form

By using a Lie algebra G and its loop algebra G ∼ , a soliton hierarchy of evolution equations is derived from which the well-known Gerdjikov–Ivanov (GI) hierarchy with two potential functions is obtained. With the help of different choices of the modified terms, two expanding integrable systems are produced, one of them has complex Hamiltonian structure. Again by making a linear transformation between G and itself, a complex Lie algebra H is obtained, the corresponding loop algebra H ∼ is defined for which a generalized GI hierarchy with self-consistent sources and its expanding integrable model are worked out, respectively.