Beyond the symmetry: Exploring the nonlinear dynamics of 1D and 2D 𝒫𝒯-symmetric sinusoidal systems

This work investigates the evolution of nonlinear eigenmodes in one-dimensional (1D) and two-dimensional (2D) optical systems with ([Formula: see text])-symmetric sinusoidal complex potentials, which supports real eigenvalue spectra up to the symmetry-breaking threshold. Numerical results show that in 1D and 2D systems, when nonlinearity increases, the threshold potential decreases. However, a deeper real potential increases the threshold in 1D, while it destabilizes the 2D optical beam as illustrated in the stability charts. Below this threshold, eigenfunctions exhibit symmetric behavior with smooth phase variations and no abrupt changes in energy flux. In contrast, the broken ([Formula: see text])-symmetric phase results in asymmetric eigenfunctions, indicating the instability. Stability analysis using Bogoliubov–de Gennes (BDG) matrix method and the propagation dynamics confirmed that eigenstates remain stable under small perturbations below the threshold but lose stability once the imaginary potential exceeds the threshold. Understanding the behavior of eigenmodes in ([Formula: see text])-symmetric potentials enables advanced control over beam profiles, potentially benefiting high-precision sensors, lasers, and optical trapping.