Parity–time ( PT ) symmetric lattices have been widely studied in controlling the flow of waves, and recently, moiré superlattices, connecting the periodic and non-periodic potentials, have been introduced for exploring unconventional physical properties in physics, while the combination of both and nonlinear waves therein remains unclear. Here, we report a theoretical survey of nonlinear wave localizations in PT symmetric moiré optical lattices, with the aim of revealing localized gap modes of different types and their stabilization mechanism. We uncover the formation, properties, and dynamics of fundamental and higher-order gap solitons as well as vortical ones with topological charge, all residing in the finite bandgaps of the underlying linear-Bloch wave spectrum. The stability regions of localized gap modes are inspected in two numerical ways: linear-stability analysis and direct perturbance simulations. Our results provide an insightful understanding of soliton physics in combined versatile platforms of PT symmetric systems and moiré patterns.

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Solitons in the fractional space, supported by lattice potentials, have recently attracted much interest. The limit of deep 1D and 2D lattices in this system is considered, featuring finite bandgaps separated by nearly flat Bloch bands. Such spectra are also a subject of great interest in current studies. The existence, shapes, and stability of various localized modes, including fundamental gap and vortex solitons, are investigated by means of numerical methods; some results are also obtained with the help of analytical approximations. In particular, the 1D and 2D gap solitons, belonging to the first and second finite bandgaps, are tightly confined around a single cell of the deep lattice. Vortex gap solitons are constructed as four-peak "squares" and "rhombuses" with imprinted winding number S = 1. Stability of the solitons is explored by means of the linearization and verified by direct simulations.