The study of solitons in nonlinear optical lattices has attracted significant theoretical and experimental interest in recent years. Nonlinear optical lattices, characterized by localized nonlinear intensities and spatially periodic modulation, demonstrate distinctive photonic properties. In particular, parity-time (PT) symmetric nonlinear optical lattices support stable discrete solitons, achievable through careful modulation of nonlinear gain and loss in specifically designed waveguides. Building upon these advances, this research investigates the integration of nonlinear lattices with linear Hermitian lattices to examine their effects on soliton optical transmission properties in nonlinear-Hermitian Moiré photonic crystal lattices under non-Hermitian conditions. Through the development of a nonlinear Schr?dinger equation for hybrid linear-nonlinear Moiré lattices and numerical simulations, this study examines bandgap soliton excitation and stability regimes. The research specifically addresses how nonlinear lattice parameter modifications, including depth, gain-loss degree, and lattice period, affect soliton stability domains, presenting findings through power-profile analysis of stable solitons. This investigation provides essential understanding for the design of photonic devices that control nonlinear soliton dynamics in non-Hermitian lattice systems.

The nonlinear Schr?dinger equation (NLSE) governs optical soliton transmission. Building upon this equation, we establish a physical model for soliton propagation in non-Hermitian Moiré photonic crystals. Through the application of the squared-operator iteration method (SOM) and its modified version (MSOM), precise soliton solutions are obtained within the bandgap structure. The stability analysis involves introducing small perturbations to the exact solutions and linearizing the NLSE, converting the stability analysis into an eigenvalue problem. The associated eigenoperator is solved using the Fourier-collocation method, where eigenmatrices are constructed from Fourier basis components to determine the linear stability spectrum of solitons. The eigenvalue spectrum analysis provides understanding of soliton transmission stability characteristics. Based on these stability findings, the split-step Fourier (SSF) method simulates soliton propagation. This method alternates between linear propagation and nonlinear interaction processing, facilitating systematic investigation of soliton transmission properties and inter-soliton interactions within the lattice. The band structure of one-dimensional non-Hermitian Moiré photonic crystals is computed using the plane-wave expansion (PWE) method. This calculation specifically addresses the linear band structure, excluding nonlinear effects. The results demonstrate a significant correlation between bandgap characteristics and stable soliton transmission thresholds.

This investigation models the nonlinear coefficient in the nonlinear Schr?dinger equation as a periodic function, establishing a linear lattice system with identical periodicity to the nonlinear Moiré lattice. With T=3 as an example (Fig. 2), increasing nonlinear lattice depth diminishes soliton power and narrows the stability region. This phenomenon stems from the interaction between self-focusing effects and gain-loss modulation, where the imaginary component of the gain-loss coefficient predominates, compelling the system to reduce soliton power. At a lattice period of 6 (Fig. 4), bandgap solitons maintain stable propagation in the semi-infinite band, independent of lattice depth variations. However, at period of 2 (Fig. 5), the soliton-lattice coupling equilibrium destabilizes due to enhanced localized gain-loss modulation, preventing stable bandgap solitons. The analysis indicates that lattice period must correspond to the soliton coherence length scale for stable propagation through periodic nonlinear modulation. The research examines soliton stability under periodic mismatch between linear (period T_L=3) and nonlinear lattices. At nonlinear lattice period of 6, the system approximates constant-coefficient behavior, facilitating stable bandgap soliton propagation with stability regions unaffected by lattice depth (Fig. 7). Increased gain-loss coefficient expands instability zones while maintaining power curves (Fig. 9). Conversely, at period of 2, intensified spatial modulation constrains stability regions (Fig. 8). Substantial gain-loss modulation (gain-loss coefficient ω₁≥0.30, Fig. 10) or sub-threshold periods disrupt soliton-lattice coupling balance, destabilizing solitons. The findings underscore lattice period matching: stable solitons require nonlinear periods above a critical threshold for quasi-constant effects, while shorter periods induce destabilizing localized nonlinear modulation.

This research establishes a composite lattice system incorporating linear and nonlinear components, governed by the nonlinear Schr?dinger equation. Through comprehensive analysis, the study examines soliton excitation and stability characteristics across various parameters, including lattice periodicity, nonlinear lattice depth, and gain-loss coefficient. The findings demonstrate that increased nonlinear lattice depth induces exponential power attenuation in solitons and substantially reduces the stable transmission window. Conversely, gain-loss coefficient variations minimally affect soliton power and stability domain. The analysis confirms that nonlinear lattice depth primarily determines soliton properties, with regulatory efficiency exceeding that of gain-loss modulation by two orders of magnitude.