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.
Highlights
The study of dispersive media exhibiting a nonlocal nonlinear response is a subject that is enjoying increasing attention over the past few years [1,2,3]
The results suggest the generic relevance of rogue waves and related structures, as well as periodic solutions, in nonlocal nonlinear Schrödinger (NLS) models
We considered a variety of wave structures existing in the local NLS equation and extended them to the realm of a generic nonlocal NLS model
Summary
The study of dispersive media exhibiting a nonlocal nonlinear response is a subject that is enjoying increasing attention over the past few years [1,2,3] This is mainly due to the fact that relevant models and their solutions, especially of the nonlinear Schrödinger (NLS) variety, emerge in a wide range of physical contexts. A key advantage of such a methodology is that it does not hinge in any critical way on integrability and, starting from the integrable limit it can be used in a variety of nonintegrable settings, such as the nonlocal one that we consider here It is this tool that will permit us to converge to the rogue wave (in the form of the Peregrine soliton) for a range of parameter values of the nonlocality parameter, referred to as ν below in our model.
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