The statistical evolution of ensembles of random, weakly interacting waves is governed by wave kinetic equations (WKEs). 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 Schrödinger–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 localizing 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 WKE. Using the SLAM, we study the stationary spectra of Schrödinger–Helmholtz wave turbulence, and characterize 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.
Bibliographical noteCopyright © 2023 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License https://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. Funding Information: This work was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska–Curie grant agreement no. 823937 for the RISE project HALT, and by the Simons Foundation Collaboration grant Wave Turbulence (Award ID 651471). J.L. and J.S. are supported by the Leverhulme Trust Project grant no. RPG-2021-014. Publisher Copyright: © 2023 The Authors.
- model reduction
- nonlinear optics
- turbulent cascades
- wave turbulence