Fundamental rogue waves and their superpositions in nonlinear integrable systems

27 pags., 8 figs.We review the large variety of exact rogue wave solutions of the nonlinear Schrödinger equation (NLSE) and its extensions. We consider fundamental rogue waves, as well as the hierarchy of higher-order rogue wave solutions that are nonlinear superpositions of fundamental ones. The fundamental rogue wave solution of the NLSE is known as the Peregrine solution. The fundamental rogue wave solutions of the NLSE extensions (including infinite extensions) can change the Peregrine shape significantly. Higher-order solutions emerge in the form of spatiotemporal patterns with a number of free parameters that control them. Rogue wave solutions are as universal as soliton or breather solutions.The authors acknowledge the support of the Volkswagen Foundation. The work of JMSC was supported by MINECO under contract TEC2015-71127-C2-1-R, and by Nonlinear Guided Wave Optics 10-22 CAM under contract S2013/MIT-2790. NA and AA acknowledge the support of the Australian Research Council (Discovery Project numbers DP140100265 and DP150102057).Peer Reviewe