Weak and strong turbulence in self-focusing and defocusing media

TL;DR

This paper investigates the differences between weak and strong turbulence in self-focusing and defocusing media, revealing how nonlinear corrections influence spectral behavior and deriving a universal large-N kinetic equation applicable across turbulence regimes.

Contribution

It introduces a large-N kinetic model that describes both weak and strong turbulence regimes, highlighting distinct universal behaviors in focusing and defocusing media.

Findings

One-loop correction causes spectral steepening in defocusing media.

Large-N kinetic equation captures turbulence at all scales.

Universal spectra are identified for strong turbulence in both media.

Abstract

While the focusing and defocusing Nonlinear Schrodinger Equations have similar behavior in the weak turbulence regime, they must differ dramatically in the strong turbulence regime. Here, we show that this difference is already present at next-to-leading order in the nonlinearity in the weak turbulence regime: The one-loop correction to the interaction vertex suppresses repulsion (like screening in electrodynamics), leading to a steeper spectrum in the defocusing case. In contrast, attraction enhancement (like antiscreening in chromodynamics) makes the spectrum less steep in the focusing case. To describe strong turbulence, we consider a vector model in the limit of a large number of components. A large-N kinetic equation, valid at all scales, can be derived analytically. It has an inverse-cascade solution whose two asymptotics, at high and low wavenumbers, describe weak and strong turbulence, respectively. We find two forms of universality in the strong turbulence spectrum: in focusing media it is independent of the flux magnitude, while in defocusing media it is independent of the bare coupling constant, with the largest scale appearing instead.