Spectral curves for the rogue waves

TL;DR

This paper investigates the spectral curves associated with rational, quasi-rational, and multi-phase solutions of AKNS hierarchy equations, linking them to rogue wave phenomena in optics and hydrodynamics, and addresses gaps in existing literature.

Contribution

It provides a detailed analysis of spectral curves for various solutions of AKNS hierarchy, expanding understanding of their role in rogue wave modeling.

Findings

Spectral curves for rational and quasi-rational solutions are characterized.

Multi-phase solutions' spectral curves are determined.

The study highlights the importance of spectral curves in rogue wave modeling.

Abstract

Here we find the spectral curves, corresponding to the known rational or quasi-rational solutions of AKNS hierarchy equations, ultimately connected with the modeling of the rogue waves events in the optical waveguides and in hydrodynamics. We also determine spectral curves for the multi-phase trigonometric, hyperbolic and elliptic solutions for the same hierarchy. It seams that the nature of the related spectral curves was not sufficiently discussed in existing literature.