Integrable decompositions for the (2 + 1)-dimensional multi-component Ablowitz–Kaup–Newell–Segur hierarchy and their applications

This paper investigates integrable decompositions of the (2 + 1)-dimensional multi-component Ablowitz-Kaup-Newell-Segur (AKNS in brief) hierarchy. By utilizing recursive relations and symmetric reductions, it is demonstrated that the (n2 − n1 + 1)-flow of the (2 + 1)-dimensional coupled multi-component AKNS hierarchy can be decomposed into the corresponding n1-flow and n2-flow of the coupled multi-component AKNS hierarchy. Specifically, except for two specific scenarios, the (n2 − n1 + 1)-flow of the (2 + 1)-dimensional reduced multi-component AKNS hierarchy can similarly be decomposed into the corresponding n1-flow and n2-flow of the reduced multi-component AKNS hierarchy. Through the application of these integrable decompositions and Darboux transformation techniques, multiple solitons for the standard focusing multi-component “breaking soliton” equations, as well as singular, exponential, and rational solitons for the nonlocal defocusing multi-component “breaking soliton” equations, are systematically presented. Furthermore, the elastic interactions and dynamical behaviors among these soliton solutions are thoroughly investigated without loss of generality.