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

TL;DR

This paper introduces a new class of higher-dimensional loop algebras to generate and analyze integrable hierarchies, including nonisospectral Schr"odinger equations, with applications in stochastic systems.

Contribution

It constructs higher-dimensional column-vector loop algebras and develops a method to generate diverse integrable hierarchies, extending classical equations and their coupling structures.

Findings

Derived a generalized nonisospectral Schr"odinger hierarchy.

Reduced hierarchies to classical equations like heat and Fokker-Planck.

Discussed Hamiltonian structures using quadratic form trace identity.

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.