Inverse scattering transform analysis of rogue waves using local periodization procedure

TL;DR

This paper introduces a novel inverse scattering transform-based method to classify rogue waves in the nonlinear Schrödinger equation by isolating local structures within incoherent wave trains, extending existing classifications.

Contribution

A new local periodization approach for the inverse scattering transform enables classification of rogue waves beyond standard breathers, incorporating general nonlinear modes.

Findings

Extended classification of rogue waves including nonlinear modes

Successful numerical implementation of local periodization for IST

Identification of new rogue wave prototypes

Abstract

The nonlinear Schr\"odinger equation (NLSE) stands out as the dispersive nonlinear partial differential equation that plays a prominent role in the modeling and understanding of the wave phenomena relevant to many fields of nonlinear physics. The question of random input problems in the one-dimensional and integrable NLSE enters within the framework of integrable turbulence, and the specific question of the formation of rogue waves (RWs) has been recently extensively studied in this context. The determination of exact analytic solutions of the focusing 1D-NLSE prototyping RW events of statistical relevance is now considered as the problem of central importance. Here we address this question from the perspective of the inverse scattering transform (IST) method that relies on the integrable nature of the wave equation. We develop a conceptually new approach to the RW classification in which appropriate, locally coherent structures are specifically isolated from a globally incoherent wave train to be subsequently analyzed by implementing a numerical IST procedure relying on a spatial periodization of the object under consideration. Using this approach we extend the existing classifications of the prototypes of RWs from standard breathers and their collisions to more general nonlinear modes characterized by their nonlinear spectra.