Rogue periodic waves of the focusing NLS equation

TL;DR

This paper constructs explicit rogue wave solutions on periodic backgrounds for the focusing NLS equation, revealing their instability, properties, and relation to classical rogue waves, with potential implications for understanding extreme wave phenomena.

Contribution

It introduces new explicit rogue wave solutions on periodic backgrounds using Darboux transformations, extending classical rogue wave models to periodic settings.

Findings

Rogue periodic waves are modulationally unstable.

Explicit solutions are derived using Zakharov-Shabat eigenfunctions.

Magnification factors depend on wave amplitude.

Abstract

Rogue waves on the periodic background are considered for the nonlinear Schrodinger (NLS) equation in the focusing case. The two periodic wave solutions are expressed by the Jacobian elliptic functions dn and cn. Both periodic waves are modulationally unstable with respect to long-wave perturbations. Exact solutions for the rogue waves on the periodic background are constructed by using the explicit expressions for the periodic eigenfunctions of the Zakharov-Shabat spectral problem and the Darboux transformations. These exact solutions labeled as rogue periodic waves generalize the classical rogue wave (the so-called Peregrine's breather). The magnification factor of the rogue periodic waves is computed as a function of the wave amplitude (the elliptic modulus). Rogue periodic waves constructed here are compared with the rogue wave patterns obtained numerically in recent publications.