Continuous limit, position controllable singular rogue wave and periodic wave solutions for a discrete reverse-time nonlocal coupled Ablowitz–Ladik equation

The discrete Ablowitz–Ladik (AL) equation is the discrete version of nonlinear Schrödinger equation, which may have potential physical applications in nonlinear optics, polaron motion and anharmonic lattice dynamics. In this paper, a discrete reverse-time nonlocal coupled AL equation is first proposed and studied. First of all, we correspond this new discrete reverse-time nonlocal equation to continuous nonlocal coupled equation by use of the continuous limit technique. Second, we build the generalized [Formula: see text]-fold Darboux transformation for this new discrete equation. As an application, we obtain some novel position controllable nonlocal singular rogue wave (RW) and period wave solutions on constant seed backgrounds, whose structures and positions are controlled by some special parameters. Moreover, we also study dynamical behaviors of some RW solutions via numerical simulations and large asymptotic analysis. These new results and phenomena may be helpful to comprehend some physical phenomena.