Derivation of Expanded Isospectral-Nonisospectral Integrable Hierarchies via the Column-vector Loop Algebra

A scheme for generating nonisospectral integrable hierarchies is introduced. Based on the method, we deduce a nonisospectral hierarchy of soliton equations by considering a linear spectral problem. It follows that the corresponding expanded isospectral and nonisospectral integrable hierarchies are deduced based on a 6 dimensional complex linear space \(\widetilde{\mathbb{C}}^{6}\) $${\tilde{\mathbb{C}}}^6$$ . By reducing these integrable hierarchies, we obtain the expanded isospectral and nonisospectral derivative nonlinear Schrödinger equation. By using the trace identity, the bi-Hamiltonian structure of these two hierarchies are also obtained. Moreover, some symmetries and conserved quantities of the resulting hierarchy are discussed.