We present the extreme rogue wave dynamics that originate from the three-wave resonant interaction equations, which describe the transient stimulated scattering processes, in cubic media, and the parametric frequency conversion processes, in phase-matched dispersive quadratic media. Explicit theoretical fundamental and second-order rogue wave solutions are presented and characterized. The link between the existence of rogue wave solutions and the presence of baseband modulation instability is highlighted. We also present numerical simulations that confirm the robustness and possible manifestation of these rogue waves for various parameters and initial conditions.

This work was supported by the Italian Ministry of University and Research (MIUR, Project No. 2012BFNWZ2), by the National Natural Science Foundation of China (Grants No. 11174050 and No. 11474051), and by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 691051. Ph. G. acknowledges funding from CEFIPRA/ IFCPAR (project 5104-2), and the LABEX ACTION (ANR-11-LABX-01-01). The work of J.M.S.C. was supported by MINECO under contract TEC2015-71127- C2-1-R, by C.A.M. under contract S2013/MIT-2790, and by the Volkswagen Foundation.

18 pags., 6 figs.

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