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

Quasi-Gramian Solution of a Noncommutative Extension of the Higher-Order Nonlinear Schr\"odinger Equation