General rogue waves of infinite order: Exact properties, asymptotic behavior, and effective numerical computation

TL;DR

This paper thoroughly analyzes infinite-order rogue wave solutions of the focusing nonlinear Schrödinger equation, detailing their properties, asymptotic behavior, and providing numerical methods for their accurate computation.

Contribution

It introduces a comprehensive framework for understanding and computing infinite-order rogue waves, including their exact properties and asymptotic analysis.

Findings

Solutions are in L^2(R) spatially but decay slowly over time.

Established exact and asymptotic properties of these rogue waves.

Developed computational tools for accurate numerical evaluation.

Abstract

This paper is devoted to a comprehensive analysis of a family of solutions of the focusing nonlinear Schr\"odinger equation called general rogue waves of infinite order. These solutions have recently been shown to describe various limit processes involving large-amplitude waves, and they have also appeared in some physical models not directly connected with nonlinear Schr\"odinger equations. We establish the following key property of these solutions: they are all in $L^2(\mathbb{R})$ with respect to the spatial variable but they exhibit anomalously slow temporal decay. In this paper we define general rogue waves of infinite order, establish their basic exact and asymptotic properties, and provide computational tools for calculating them accurately.