Rogue waves statistics in the framework of one-dimensional Generalized Nonlinear Schrodinger Equation

TL;DR

This study investigates rogue wave statistics in one-dimensional NLS-like equations, revealing how small perturbations and higher-order interactions influence spectral properties, correlation functions, and wave amplitude distributions.

Contribution

It provides a comparative analysis of classical, nonintegrable, and generalized NLS equations, highlighting the effects of additional terms on rogue wave statistics and probability density functions.

Findings

Classical NLS with condensate shows non-decaying correlations and breathing PDFs.

Adding damping and pumping breaks integrability, leading to Rayleigh PDFs.

Higher-order interactions increase non-Rayleigh features in wave amplitude distributions.

Abstract

We measure evolution of spectra, spatial correlation functions and probability density functions (PDFs) of waves appearance for a set of one-dimensional NLS-like equations of focusing type, namely for the classical integrable Nonlinear Schrodinger equation (1), nonintegrable NLS equation accounting for dumping (linear dissipation, two- and three-photon absorption) and pumping terms (2) and generalized NLS equation accounting for six-wave interactions, dumping and pumping terms (3). All additional terms beyond the classical NLS equation are small. As initial conditions we choose seeded by noise modulationally unstable solutions of the considered systems in the form of (a) condensate for systems (1)-(3) and (b) cnoidal wave for the classical NLS equation (1). We observe 'strange' results for the classical NLS equation (1) with condensate initial condition including peak at zeroth harmonic in averaged over ensemble spectra, non-decaying spacial correlation functions and a 'breathing' region on the PDFs for medium waves amplitudes where frequency of waves appearance oscillates with time, while the far-tails of the PDFs remain Rayleigh ones. Addition of small dumping and pumping terms in model (2) breaks integrability that results in absence of the peak at zeroth harmonic in spectra, spacial correlation functions decaying to zero level and strictly Rayleigh PDFs for waves amplitudes. For the classical NLS equation (1) with cnoidal wave initial condition PDFs turn out to be significantly different from Rayleigh ones with 'fat tails' in the region of large amplitudes where higher waves appear more frequently, while for generalized NLS equation with six-wave interactions, dumping and pumping terms (3) we demonstrate absence of non-Rayleigh addition to the PDFs for zeroth six-wave interactions coefficient and increase of non-Rayleigh addition with six-wave interactions term.