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The statistical evolution of ensembles of random, weakly-interacting waves isgoverned by wave kinetic equations. To simplify the analysis, one frequentlyworks with reduced differential models of the wave kinetics. However, theconditions for deriving such reduced models are seldom justifiedself-consistently. Here, we derive a reduced model for the wave kinetics of theSchr{\"o}dinger-Helmholtz equations in two spatial dimensions, which constitutea model for the dynamics of light in a spatially-nonlocal, nonlinear opticalmedium. This model has the property of sharply localising the frequencies ofthe interacting waves into two pairs, allowing for a rigorous andself-consistent derivation of what we term the semilocal approximation model(SLAM) of the wave kinetic equation. Using the SLAM, we study the stationaryspectra of Schr{\"o}dinger-Helmholtz wave turbulence, and characterise thespectra that carry energy downscale, and waveaction upscale, in aforced-dissipated setup. The latter involves a nonlocal transfer of waveaction,in which waves at the forcing scale mediate the interactions of waves at everylarger scale. This is in contrast to the energy cascade, which involves localscale-by-scale interactions, familiar from other wave turbulent systems andfrom classical hydrodynamical turbulence.

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