  The nonlinear Schr{\"o}odinger (NLS) equation, which incorporates
higher-order dispersive terms, is widely employed in the theoretical analysis
of various physical phenomena. In this study, we explore the non-commutative
extension of the higher-order NLS equation (HNLS). We treat real or
complex-valued functions, such as g1 = g1(x, t) and g2 = g2(x, t), as
non-commutative, and employ the Lax pair associated with the evolution equation
as in the commutation case. We derive the quasi-Gramian solution of the system
by employing a binary Darboux transformation (DT). Moreover, the solution can
be used to study the stability of plane waves and to understand the generation
of periodic patterns in the context of modulational instability.
