The evolution of Cos−Gaussian beams in periodic potential optical lattices is theoretically and numerically investigated. By theoretical analysis, a breathing soliton solution of the Gross–Pitaevskii equation with periodic potential is obtained, and the period of the breathing soliton is solved. In addition, the evolution of Cos−Gaussian beams in periodic potential optical lattices is numerically simulated. It is found that breathing solitons generate by appropriately choosing initial medium and beam parameters. Firstly, the effects of the initial parameters of Cos−Gaussian beams (initial phase and width) on its initial waveform and the propagation characteristics of breathing soliton are discussed in detail. Then, the influence of the initial parameters (modulation intensity and modulation frequency) of a photonic lattice on the propagation characteristics of breathing solitons is investigated. Finally, the effects of modulation intensity and modulation frequency on the width and period of the breathing soliton are analyzed. The results show that the number of breathing solitons is manipulated by controlling the initial parameters of Cos−Gaussian beams. The period and width of a breathing soliton are controlled by manipulating the initial parameters of a periodic photonic lattice. The results provide some theoretical basis for the generation and manipulation of breathing solitons.

The propagation characteristics of Airy beams in an inhomogeneous medium with periodic potential are studied theoretically and numerically. The Gross–Pitaevskii equation was solved with periodic potential using the separating variables method, and a breathing soliton solution and the breathing period were obtained. Further, the propagation properties of an Airy beam, and the interaction between two Airy beams while considering the medium parameters and beam parameters were numerically simulated in detail. First, we discuss the influence of the initial medium parameters (modulation intensity P and modulation frequency ω) on the propagation characteristics. Then, we investigate the effect of the initial beam parameters (initial chirp C and position x₀) on the propagation characteristics. Lastly, the interaction of two Airy beams with opposite spatial positions for different phase φ, amplitude A, and initial interval x₀ is analyzed. The breathing period and central position of the breathing solitons could be controlled by changing the initial medium parameters. By varying the initial beam parameters, the deflection direction and size, and the maximal intensity of the breathing solitons were manipulated. The breathing solitons of different bound states were formed by changing the phase φ, amplitude A, and initial interval x₀ of two Airy beams. The results provide a theoretical basis for the propagation and manipulation of Airy beams.

The Airyprime beam, due to its adjustable focusing ability and controllable orbital angular momentum, has attracted significant attention in fields such as free-space optical communication and particle trapping. However, systematic studies on the propagation behavior of oscillating solitons in PT-symmetric optical lattices remain scarce, particularly regarding their formation mechanisms and self-accelerating characteristics. In this study, the propagation characteristics of Airyprime beams in PT symmetric optical lattices are numerically studied using the split-step Fourier method, and the generation mechanism and control factors of oscillating solitons are analyzed. The influence of lattice parameters (such as the modulation depth P, modulation frequency w, and gain/loss distribution coefficient W₀) and beam initial characteristics (such as the truncation coefficient a) on the dynamic behavior of the beam is revealed. The results show that the initial parameters determine the propagation characteristics of the beam and the stability of the soliton. This research provides theoretical support for beam shaping, optical path design, and nonlinear optical manipulation and has important application value.