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We demonstrate a possibility to make rogue waves (RWs) in the form of thePeregrine soliton (PS) and Kuznetsov-Ma breathers (KMBs) effectively stableobjects, with the help of properly defined dispersion or nonlinearitymanagement applied to the continuous-wave (CW) background supporting the RWs.In particular, it is found that either management scheme, if applied along thelongitudinal coordinate, making the underlying nonlinear Schr\"odinger equation(NLSE) selfdefocusing in the course of disappearance of the PS, indeedstabilizes the global solution with respect to the modulational instability ofthe background. In the process, additional excitations are generated, namely,dispersive shock waves and, in some cases, also a pair of slowly separatingdark solitons. Further, the nonlinearity-management format, which makes theNLSE defocusing outside of a finite domain in the transverse direction, enablesthe stabilization of the KMBs, in the form of confined oscillating states. Onthe other hand, a nonlinearity-management format applied periodically along thepropagation direction, creates expanding patterns featuring multiplication ofKMBs through their cascading fission.

Stabilization of the Peregrine soliton and Kuznetsov-Ma breathers by means of nonlinearity and dispersion management