An integrable system of two-component nonlinear Ablowitz-Ladik equations is used to construct complex rogue-wave (RW) solutions in an explicit form. First, the modulational instability of continuous waves is studied in the system. Then, new higher-order discrete two-component RW solutions of the system are found by means of a newly derived discrete version of a generalized Darboux transformation. Finally, the perturbed evolution of these RW states is explored in terms of systematic simulations, which demonstrates that tightly and loosely bound RWs are, respectively, nearly stable and strongly unstable solutions.

In this work, the (2 + 1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation is investigated. Hirota's bilinear method is used to determine the N-soliton solutions for this equation, from which the M-lump solutions are obtained by using long wave limit when N is even (i.e., N = 2M). Then, taking N = 5 as an example, we discuss some novel mixed lump-soliton and lump-soliton-breather solutions by using long wave limit and choosing special conjugate complex parameters from the five-soliton solution. Figures are plotted to reveal the dynamical features of such obtained lump and mixed interaction solutions. These results may be useful for understanding the propagation phenomena of nonlinear localized waves.

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