The quasi-Gramian solution of a non-commutative extension of the higher-order nonlinear Schrödinger equation

Abstract

The nonlinear Schr{\"o}dinger (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 (HNLS) equation. We treat real or complex-valued functions, such as $g_1=g_1 (x,\;t)$ and $g_2= g_2 (x ,\;t)$ 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). \textcolor[rgb]{0.50,0.00,0.00}{The soliton solutions are presented explicitly within framework of quasideterminants. In order to visually understand the dynamics and solutions in the given example, we also provide simulations illustrating the associated profiles.} 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.