# Weak and strong optical turbulence in the Schrödinger Helmholtz Equation

Tuesday 03, 17:15

Jonathan Skipp

Aston university

In this talk I will give an overview of results that Sergey has led us to, on the Schrödinger Helmholtz Equation (SHE). The SHE is a model that describes nonlinear optics beyond the Kerr regime (it also has applications to cosmology, modelling dark matter as an ultra low-mass boson). As such, it is an extension of the familiar nonlinear Schrödinger equation, incorporating a spatially nonlocal self-potential. In 1D, we have examined how the non-integrability of the SHE leads to interesting dynamics of solitons: interactions with weakly nonlinear waves, generation of bound soliton states, and soliton mergers. I will show how a tool borrowed from integrable systems theory, the direct scattering transform, can be used to help characterise these dynamics, even when the system is quite far from being integrable. In 2D I return to wave turbulence, and will summarise how the SHE has a nonlinearity that allows for a reduction of the collision integral into a much simpler form, and the cascade solutions found. This is the first time in the wave turbulence literature that a reduced kinetic model can be obtained fully analytically.

## References:

- [1] J. Laurie, U. Bortolozzo, S. Nazarenko, and S. Residori, Phys. Rep., 514, 4, 121, (2012)
- [2] C. Colléaux, J. Skipp, J. Laurie, and S. Nazarenko, in preparation.
- [3] J. Skipp, V. L’vov, and S. Nazarenko, Physical Review A, 102, 4, 043318, (2020)
- [4] J. Skipp, J. Laurie, and S. Nazarenko, Proc. R. Soc. A, 479, 2275, 20230162, (2023)