Spatiotemporal dynamics of necklace ring solitons in the whispering-gallery-mode optical microcavity

Dynamical characteristics of spatiotemporal necklace ring solitons in the whispering gallery mode optical microcavity is investigated based on the (3 + 1)-dimensional variable-coefficient Lugiato-Lefever equation. An analytical solution for the case without external pumping is obtained. The effects of the azimuthal periodic modulation, noise perturbation, time-varying excitation, topological charge and diffraction parameters on the stability, spatial structure and energy distribution of soliton are analyzed both in the absence and presence of external pumping. The azimuthal periodic modulation can precisely control the number of circumferential lobes in the soliton, with the number of lobes strictly matching the modulation period. The noise perturbation induces only local structural fluctuations, without disrupting the global periodicity. Low-intensity time-varying excitation, in combination with the periodic modulation, can maintain the long-term stability of solitons. The topological charge reconstructs the energy distribution via the phase winding effect, causing the symmetry distortion without altering the maximum amplitude. Linear stability analysis confirms that the diffraction parameter influences the soliton stability by modulating the nonlinear coupling strength, and the smaller diffraction coefficient is more favorable for maintaining the stable evolution. These findings provide a theoretical foundation for controlling the spatiotemporal structure of light fields and the design of nonlinear modes in the whispering-gallery mode microcavity.