Analysis, Dynamics, and Applications Seminar

The dynamical and statistical behaviour of random weakly-interacting waves is responsible for many important physical effects across applications ranging from quantum to classical and to astrophysical scales. Such behaviour is described by the theory of weak wave turbulence. A characteristic prediction of weak wave turbulence theory is that of nonequilibrium stationary states, in which dynamical invariants cascade through scales. Sometimes, similar to 2D classical turbulence, cascades are dual, with one invariant cascading to small scales, and the other invariant towards large scales. The latter cascade often leads to accumulation of the turbulence spectrum near the largest scale of the system, which is analogous to Bose-Einstein condensation. Condensation via the dual cascade is therefore a universal mechanism by which large-scale structure may develop in a system from random waves, realising a scenario of order emerging from chaos.

In this talk I will present the wave turbulence description of the dual cascade process that precedes structure formation in the Schrödinger-Helmholtz equations (SHE). In 3D, the SHE describes the large-scale behaviour of so-called “fuzzy dark matter” in which cosmological dark matter particles are hypothesised to be an ultralight (m ∼ 10−²²eV) fundamental scalar boson. The condensate in this case can be interpreted as a galactic dark matter halo. In 2D, the SHE describes spatially-nonlocal nonlinear optics, in which the inverse cascade results in solitonic structures that are analogues of galactic haloes in tabletop experiments. The analysis of the dual cascade is made possible by the development of reduced models of the wave kinetic equation of the SHE [1, 2]. In wave turbulence, such reduced models are often constructed in a somewhat ad-hoc manner. The SHE is particularly interesting as it is the first system studied in wave turbulence where a reduced model of the wave kinetics can be derived rigorously [2]. The dual cascade spectra can then be found with relative ease, as I will demonstrate in this talk.

References

[1] Jonathan Skipp, Victor L’vov, and Sergey Nazarenko, Wave turbulence in self-gravitating Bose gases and nonlocal nonlinear optics, Phys. Rev. A, 102, 043318 (2020)

[2] Jonathan Skipp, Jason Laurie, and Sergey Nazarenko, An effective semilocal model for wave turbulence in 2D nonlinear optics, to appear on arXiv (2023)