Transdimensional Approach for Studying Multidimensional Cavity Solitons

Optical solitons, which remain localized during propagation, arise in 1D, 2D, and 3D contexts. In 1D, temporal solitons result from the balance between Kerr nonlinearity and dispersion [1], as confirmed by the experiments in fiber ring resonators [2] and microresonators [3], underpinning optical frequency comb generation. In 2D, cavity solitons appear as spatially localized monochromatic waves [4] and are also predicted in spatiotemporal regimes [5],[6]. However, in higher dimensions, solitons are susceptible to numerous instabilities, such as wave collapse [7], prompting fundamental questions about how their behavior evolves with dimensionality. To address these challenges, we introduced a dimensional reduction approach [8], which simplifies multidimensional stability analysis by mapping them onto an effective 1D model. This technique has already been employed to investigate solitons in passively driven Kerr cavities with parabolic potentials, both in 1D [10] and 3D [9].