The integrable nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential [M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 110, 064105 (2013)] is investigated, which is an integrable extension of the standard nonlinear Schrödinger equation. Its novel higher-order rational solitons are found using the nonlocal version of the generalized perturbation $(1,N-1)$ -fold Darboux transformation. These rational solitons illustrate abundant wave structures for the distinct choices of parameters (e.g., the strong and weak interactions of bright and dark rational solitons). Moreover, we also explore the dynamical behaviors of these higher-order rational solitons with some small noises on the basis of numerical simulations.

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.

The higher order discrete rogue waves (RWs) of the integrable discrete Ablowitz-Ladik equation are reported using a novel discrete version of generalized perturbation Darboux transformation. The dynamical behaviors of strong and weak interactions of these RWs are analytically and numerically discussed, which exhibit the abundant wave structures. We numerically show that a small noise has the weaker effect on strong-interaction RWs than weak-interaction RWs, whose main reason may be related to main energy distributions of RWs. The interaction of two first-order RWs is shown to be non-elastic. Moreover, we find that the maximal number (S_max) of the possibly split first-order ones of higher order RWs is related to the number (P_max) of peak points of their strongest-interaction cases, that is, S_max = (P_max + 1)/2. The results will excite to further understand the discrete RW phenomena in nonlinear optics and relevant fields.

We investigate the integrable repulsive AB system and construct its Darboux transformation using the loop group method. The associated N-fold Darboux transformation is found in terms of simple determinants. Moreover, we derive multi-dark-dark solitons of the repulsive AB system with a non-vanishing background through the Darboux transformation with a limit procedure. Particularly, we exhibit the one-, two-, and three-dark-dark solitons. The results will be meaningful for the study of vector multi-dark solitons in many physical systems such as geophysical fluid dynamics and nonlinear optics.