Space-time controllable Airyprime Hermite complex-variable-function Gaussian wave packets in the variable diffraction fractional system

This paper investigates the propagation characteristics of space-time Airyprime Hermite complex-variable-function Gaussian (AHCG) wave packets in the variable coefficient fractional Schrödinger equation (FSE). By adjusting the distribution coefficient, the wave packet exhibits diverse structural forms, including circular, elliptical ring, double vortex, butterfly, figure of eight, and solid circle. Cosine modulation leads to periodic expansion and contraction, while the cross phase induces periodic inversion during transmission. The Lévy index governs the magnitudes of space diffraction and time dispersion, leading to different spatial splitting and pulse stacking phenomena. Temporal chirp further modulates the wave packet's acceleration direction, velocity magnitude, lobe profile, and energy distribution in the time domain. Primary spatial chirps introduce serpentine propagation trajectories alongside periodic recovery of the spatial structure. Secondary space chirps regulate spatial energy diffusion and create a breathing state, with alternation influenced by the chirp symbol. The modulation frequency significantly impacts evolutionary period and wave packet expansion and shrinkage. Variations in cross-phase coefficients and Lévy indices affect deflection direction and diffraction patterns, leading to single, double vortex and multiple split spatial structures. The time-domain wave packet produces a different number of pulse stacks as decay factor varies. These findings highlight unique properties of AHCG wave packets in the FSE, offering novel insights for optical communications and light trapping applications.