Rogue waves and periodic solutions of a nonlocal nonlinear Schrödinger model

In the present paper, a nonlocal nonlinear Schrödinger (NLS) model is studied by means of a recent technique that identifies solutions of partial differential equations by considering them as fixed points in space-time. This methodology allows us 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 and (b) the generalization thereof in the form of the Kuznetsov-Ma breather, as well as two spatiotemporally periodic solutions in the form of elliptic functions. Importantly, all four wave forms 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.