A class of extended high-dimensional nonisospectral KdV hierarchies and symmetry

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Abstract We construct a new class of N-dimensional Lie algebras and apply them to integrable systems. In this paper, we obtain a nonisospectral KdV integrable hierarchy by introducing a nonisospectral problem. Then, a coupled nonisospectral KdV hierarchy is deduced by means of the corresponding higher-dimensional loop algebra. It follows that the K symmetries, τ symmetries, and their Lie algebra of the coupled nonisospectral KdV hierarchy are investigated. The bi-Hamiltonian structures of the both resulting hierarchies are derived by using the trace identity. Finally, we derive a multicomponent nonisospectral KdV hierarchy related to the N-dimensional loop algebra, which generalizes the coupled results to an arbitrary number of components.