Abundant discrete localized wave solutions and related dynamic analysis for the coupled Ablowitz–Ladik equation with a time-dependent coefficient

Under consideration is a new discrete coupled Ablowitz–Ladik equation with a time-dependent coefficient. Firstly, the continuum limit technique is used to correspond this discrete equation to two continuous variable coefficient equations. Secondly, the modulation instability on plane wave solutions is used to study the formation mechanism of diverse localized waves. Thirdly, the discrete generalized \((m, N-m)\) $(m,N-m)$ -fold Darboux transformation is established and extended to solve this discrete equation. Finally, as an application of the resulting generalized Darboux transformation, abundant localized wave solutions including rogue wave, period wave and mixed interaction solutions are analyzed and presented graphically. Compared with its corresponding constant coefficient discrete equation, this variable coefficient equation has more abundant rogue wave structures, and it is found that a fundamental rogue wave can be split into two fundamental rogue waves through choosing time-dependent function and different control parameters. These new results and phenomena might be helpful for understanding the interaction and propagation of optical pulses.