Fundamental rogue waves and their superpositions in nonlinear integrable systems

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 Editor acknowledges support from the Ministry of Education and Science of the Russian Federation (Grant No. 14.Y26.31.0017