Strong Collapse Turbulence in Quintic Nonlinear Schr\"odinger Equation

TL;DR

This paper investigates the dynamics of turbulence caused by quintic nonlinearity in the one-dimensional nonlinear Schrödinger equation, revealing universal statistical properties and the role of dissipation in preventing singularities.

Contribution

It introduces a detailed analysis of collapse-driven turbulence in the quintic nonlinear Schrödinger equation, highlighting universal correlation functions and the statistical nature of large amplitude fluctuations.

Findings

Collapse events form a universal spatio-temporal pattern.

The correlation function has a universal form with a specific correlation length.

Large amplitude fluctuations follow a power-law tail in their PDF.

Abstract

We consider the quintic one dimensional nonlinear Schr\"odinger equation with forcing and both linear and nonlinear dissipation. Quintic nonlinearity results in multiple collapse events randomly distributed in space and time forming forced turbulence. Without dissipation each of these collapses produces finite time singularity but dissipative terms prevents actual formation of singularity. In statistical steady state of the developed turbulence the spatial correlation function has a universal form with the correlation length determined by the modulational instability scale. The amplitude fluctuations at that scale are nearly-Gaussian while the large amplitude tail of probability density function (PDF) is strongly non-Gaussian with power-like behavior. The small amplitude nearly-Gaussian fluctuations seed formation of large collapse events. The universal spatio-temporal form of these events together with the PDF for their maximum amplitudes define the power-like tail of PDF for large amplitude fluctuations, i.e., the intermittency of strong turbulence.