Integrable decomposition for the (2+1)-dimensional AKNS hierarchy and its applications

In this paper, we are concerned with the integrable decomposition for the (2+1)-dimensional Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy. By utilizing recursive relations and symmetric reductions, we propose that the (n2 − n1 + 1)-flow of the (2+1)-dimensional AKNS hierarchy can be decomposed into the corresponding n1-flow and n2-flow of the (1+1)-dimensional AKNS hierarchy, both in the coupled and reduced cases. As an appropriate generalization, the integrable decompositions for the standard (2+1)-dimensional Heisenberg ferromagnet equation, the standard (2+1)-dimensional modified Heisenberg ferromagnet equation, and their two coupled generalizations are investigated. With no loss of generality, one-soliton solutions and their dynamic projections for the relevant gauge equivalent structures are discussed and illustrated through some figures.