An effective semilocal model for wave turbulence in 2D nonlinear optics

The statistical evolution of ensembles of random, weakly-interacting waves is governed by wave kinetic equations. To simplify the analysis, one frequently works with reduced differential models of the wave kinetics. However, the conditions for deriving such reduced models are seldom justified self-consistently. Here, we derive a reduced model for the wave kinetics of the Schr{\"o}dinger-Helmholtz equations in two spatial dimensions, which constitute a model for the dynamics of light in a spatially-nonlocal, nonlinear optical medium. This model has the property of sharply localising the frequencies of the interacting waves into two pairs, allowing for a rigorous and self-consistent derivation of what we term the semilocal approximation model (SLAM) of the wave kinetic equation. Using the SLAM, we study the stationary spectra of Schr{\"o}dinger-Helmholtz wave turbulence, and characterise the spectra that carry energy downscale, and waveaction upscale, in a forced-dissipated setup. The latter involves a nonlocal transfer of waveaction, in which waves at the forcing scale mediate the interactions of waves at every larger scale. This is in contrast to the energy cascade, which involves local scale-by-scale interactions, familiar from other wave turbulent systems and from classical hydrodynamical turbulence.