Integrable turbulence and formation of rogue waves

TL;DR

This paper investigates the nonlinear evolution of modulation instability in the focusing NLS equation, revealing the formation of integrable turbulence and analyzing its statistical properties through numerical simulations.

Contribution

It provides a detailed numerical study of integrable turbulence formation and its statistical characteristics in the focusing NLS equation, which was not comprehensively explored before.

Findings

Development of integrable turbulence from modulation instability

Statistical properties of wave amplitudes and energies analyzed

Wave-action spectrum and correlation functions characterized

Abstract

In the framework of the focusing Nonlinear Schrodinger (NLS) equation we study numerically the nonlinear stage of the modulation instability (MI) of the condensate. As expected, the development of the MI leads to formation of "integrable turbulence" [V.E. Zakharov, Turbulence in integrable systems, Stud. in Appl. Math. 122, no. 3, 219-234, (2009)]. We study the time evolution of it's major characteristics averaged across realizations of initial data - the condensate solution seeded by small random noise with fixed statistical properties. The measured quantities are: (1) wave-action spectrum and spatial correlation function, (2) the probability density function (PDF) of wave amplitudes and their momenta, and (3) kinetic and potential energies.