Exploring discrete rogue wave, hybrid wave, and their dynamics in a semi-discrete coherently coupled NLS equation featuring a 4 × 4 matrix spectral problem.

This paper delves into a semi-discrete coherently coupled nonlinear Schrödinger equation characterized by a 4×4 matrix spectral problem. Our primary objective is to explore the modulation instability theory of this equation, elucidating its formation mechanism from its plane wave solutions. Second, we aim to demonstrate that this equation can be transformed into a new continuous equation in the context of the continuous limit. Notably, utilizing the established 4×4 matrix spectral problem, we establish a discrete generalized (m,N-m)-fold Darboux transformation, from which we theoretically derive novel rogue wave and periodic wave solutions, as well as their hybrid counterparts. In particular, we obtain discrete rogue waves featuring double peaks and double troughs on a plane wave background, as well as those that exhibit only peaks and lack troughs on a zero background, both of which incorporate arbitrarily controllable position parameters. Subsequently, we graphically analyze all these innovative structures. These findings may hold potential implications for describing the optical pulse propagation in the optical fiber.