Rogue waves in multiphase solutions of the focusing NLS equation

TL;DR

This paper explores the formation of generalized rogue waves within multiphase solutions of the focusing nonlinear Schrödinger equation, providing analytical criteria for their occurrence based on the topology of real tori.

Contribution

It introduces a new analytical framework to identify and understand generalized rogue waves in finite-band fNLS solutions using the winding of real tori.

Findings

Derived an analytical criterion for generalized rogue waves

Linked rogue wave formation to the winding of real tori

Extended the understanding of rogue waves beyond standard breather solutions

Abstract

Rogue waves appearing on deep water or in optical fibres are often modelled by certain breather solutions of the focusing nonlinear Schr\"odinger (fNLS) equation which are referred to as solitons on finite background (SFBs). A more general modelling of rogue waves can be achieved via the consideration of multiphase, or finite-band, fNLS solutions of whom the standard SFBs and the structures forming due to their collisions represent particular, degenerate, cases. A generalised rogue wave notion then naturally enters as a large-amplitude localised coherent structure occurring within a finite-band fNLS solution. In this paper, we use the winding of real tori to show the mechanism of the appearance of such generalized rogue waves and derive an analytical criterion distinguishing finite-band potentials of the fNLS equation that exhibit generalised rogue waves.