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It is shown that sufficiently large periodic modulations in the coefficientsof a nonlinear Schr{\"o}dinger equation can drastically impact the spatialshape of the Peregrine soliton solutions: they can develop multiple compressionpoints of the same amplitude, rather than only a single one, as in thespatially homogeneous focusing nonlinear Schr{\"o}dinger equation. Theadditional compression points are generated in pairs forming a comb-likestructure. The number of additional pairs depends on the amplitude of themodulation but not on its wavelength, which controls their separation distance.The dynamics and characteristics of these generalized Peregrine soliton areanalytically described in the case of a completely integrable modulation. Anumerical investigation shows that their main properties persist innonintegrable situations, where no exact analytical expression of thegeneralized Peregrine soliton is available. Our predictions are in goodagreement with numerical findings for an interesting specific case of anexperimentally realizable periodically dispersion modulated photonic crystalfiber. Our results therefore pave the way for the experimental control andmanipulation of the formation of generalized Peregrine rogue waves in the wideclass of physical systems modeled by the nonlinear Schr{\"o}dinger equation.

Peregrine comb: multiple compression points for Peregrine rogue waves in periodically modulated nonlinear Schr{\"o}dinger equations