Generation of higher-dimensional isospectral-nonisospectral integrable hierarchies associated with a new class of higher-dimensional column-vector loop algebras

Abstract

We construct a new class of higher-dimensional column-vector loop algebras. Based on it, a method for generating higher-dimensional isospectral-nonisospectral integrable hierarchies is proposed. As an application, we derive a generalized nonisospectral integrable Schr\"odinger hierarchy which can be reduced to the famous derivative nonlinear Schr\"odinger equation. By using the higher-dimensional column-vector loop algebras, we obtain an expanded isospectral-nonisospectral integrable Schr\"odinger hierarchy which can be reduced to many classical and new equations, such as the expanded nonisospectral derivative nonlinear Schr\"odinger system, the heat equation, the Fokker-Plank equation which has a wide range of applications in stochastic dynamic systems. Furthermore, we deduce a ZN nonisospectral integrable Schr\"odinger hierarchy, which means that the coupling results are extended to an arbitrary number of components. Additionally, the Hamiltonian structures of these hierarchies are discussed by using the quadratic form trace identity.