A matrix second-order negative Ablowitz–Kaup–Newell–Segur flow and its Darboux transformation

Negative-order integrable flows have significant applications in areas such as fluid dynamics, nonlinear optics, and quantum field theory, particularly in the study of soliton behavior and wave interactions. In this paper, we study a matrix second-order negative flow within the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy, formulated through a Lax pair whose temporal part involves a second-order pole in the spectral parameter. This framework facilitates the analysis of shallow water wave dynamics, including the emergence of lump and soliton solutions. The corresponding Darboux transformation is constructed within the general AKNS-type framework. Starting from a seed solution, a class of explicit solutions is generated through a single application of the Darboux transformation.