Rogue Waves and Periodic Solutions of a Nonlocal Nonlinear Schr\"odinger Model

TL;DR

This paper explores solutions of a nonlocal nonlinear Schrödinger model, demonstrating how classical waveforms like rogue waves and periodic solutions can be continued into the nonlocal regime, revealing their persistence and bifurcation behaviors.

Contribution

It introduces a method to extend known local NLS solutions to the nonlocal case and analyzes their continuation and bifurcation properties.

Findings

Rogue waves and periodic solutions can be continued in nonlocal NLS models.

Periodic solutions can be extended to arbitrary nonlocalities.

Periodic solutions merge with spatially periodic structures through bifurcation.

Abstract

In the present work, a nonlocal nonlinear Schr\"odinger (NLS) model is studied by means of a recent technique that identifies solutions of partial differential equations, by considering them as fixed points in {\it space-time}. This methodology allows to perform a continuation of well-known solutions of the local NLS model to the nonlocal case. Four different examples of this type are presented, namely (a) the rogue wave in the form of the Peregrine soliton, (b) the generalization thereof in the form of the Kuznetsov-Ma breather, as well as two spatio-temporally periodic solutions in the form of elliptic functions. Importantly, all four waveforms can be continued in intervals of the parameter controlling the nonlocality of the model. The first two can be continued in a narrower interval, while the periodic ones can be extended to arbitrary nonlocalities and, in fact, present an intriguing bifurcation whereby they merge with (only) spatially periodic structures. The results suggest the generic relevance of rogue waves and related structures, as well as periodic solutions, in nonlocal NLS models.