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