An Integrable Nonlinear Schrödinger System with Symmetric Matrix Potentials and Its Binary Darboux Transformations

This study aims to propose a class of binary Darboux transformations for an integrable nonlinear Schrödinger system with two symmetric matrix potentials. The associated Lax pairs of AKNS type ensure the existence of these binary Darboux transformations, whose explicit single-step applications yield soliton solutions. Notably, the M-matrix appearing in the formulation of the binary Darboux transformations must be constructed within an extended framework, allowing for cases where eigenvalues coincide with adjoint eigenvalues, thereby encompassing generalized Darboux transformations as well.