A noncommutative negative order AKNS (NC-AKNS(-1)) equation is studied. To show the integrability of the system, we present explicitly the underlying integrable structure such as Lax pair, zero-curvature condition, an infinite sequence of conserved densities, Darboux transformation (DT) and quasideterminant soliton solutions. Moreover, the NC-AKNS(-1) equation is compared with its commutative counterpart not only on the level of nonlinear evolution equation but also for the explicit solutions.

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Higher-order nonlinear evolution equations are important for describing the wave propagation of second- and higher-order number of fields in optical fiber systems with higher-order effects. One of these equations is the coupled complex modified Korteweg–de Vries (ccmKdV) equation. In this paper, we study noncommutative (nc) generalization of ccmKdV equation. We present Darboux and binary Darboux transformations (DTs) for the nc-ccmKdV equation and then construct its Quasi-Grammian solutions. Further, single and double-hump soliton solutions of first- and second-order are given in commutative settings.