Modulational instability and discrete rogue waves with adjustable positions for a two-component higher-order Ablowitz–Ladik system associated with 4 × 4 Lax pair

An exactly solvable two-component higher-order Ablowitz–Ladik system is introduced and investigated. It can simulate the evolution of an optical field in a tightly linked waveguide array. The generalized (m,N−m)-fold Darboux transformation is used to construct two distinct types of discrete rogue waves (RWs) with adjustable positions, namely, classical and oscillating RWs, by applying two distinct Taylor expansions to solutions of the 4 × 4 Lax pair. The dynamics of strong and weak interactions of the resulting RWs are discussed analytically, and some are discussed numerically, which demonstrate luxuriant RW structures. It is shown that novel oscillating RWs with adjustable positions exhibit unique features in numbers and shapes compared to classical RWs. In particular, we find that the novel second-order RW can own three or six basic RWs, and the novel third-order RW can own six or twelve basic RWs, while first-order RWs always have only one basic RW. Except for first-order RWs, the maximum amount (Tmax) of potentially divided first-order RWs in regard to novel RWs is correlated with the maximum amount (Smax) of classical RWs, namely, Tmax=2Smax. Moreover, the numerical results show that small noises have a lesser influence on novel strong interaction RWs than weak interaction RWs, whose primary cause could be connected to major energy distributions. The findings presented in this work will contribute to a deeper comprehending of the discrete RW phenomenon in nonlinear optics and other relevant areas.