Unveiling propagation dynamics of spatiotemporal optical vortices in nonlocal photothermal media via hybrid variational approach

Spatiotemporal optical vortices (STOVs) carrying transverse orbital angular momentum have attracted significant attention in recent years. However, unveiling their propagation dynamics in realistic nonlocal photothermal media remains a major challenge, due to the coupled effects of spatial thermal diffusion, finite temporal relaxation, and dissipation-induced time-reversal symmetry breaking. Here, we establish a self-consistent dual-field analytical framework to resolve the spatiotemporal evolution of STOVs in nonlocal photothermal media. This is achieved by introducing a hybrid strategy that combines a Lagrangian-based variational treatment for the optical field with a Galerkin-weighted residual method for the thermal field, circumventing the limitations of conservative Hamiltonian formulations in dissipative systems. Applying this analytical framework, we reveal rich propagation dynamics, including soliton states, breather states, and dissipative states, of STOVs induced by spatiotemporal nonlocality. More impressively, these three states can be switched on demand by jointly manipulating spatial diffusion length and thermal relaxation time. We further evaluate the feasibility of hybrid variational approach within a certain propagation distance. The physical mechanism behind these exotic propagation dynamics of STOVs may originate from a non-zero residual phase chirp distorting the Poynting vector flow around the spatiotemporal vortex singularity. This work provides fundamental physical insights and an analytical basis for the deterministic control of structured topological light fields in complex nonlocal media.