  Optical rogue waves and its variants have been studied quite extensively in
the context of optical fiber in recent years. It has been realized that
dispersion management in optical fiber is experimentally much more feasible
compared to its nonlinear counterpart. In this work, we report Kuznetsov-Ma
(KM)-like breathers from the first three orders of rational solutions of the
nonlinear Schr\"{o}dinger equation with periodic modulation of the dispersion
coefficient along the fiber axis. The breather dynamics are then controlled by
proper choice of modulating parameters. Additionally, the evolution of new
one-peak and two-peak breather-like solutions has been displayed corresponding
to the second-order rational solution. Direct numerical simulations based on
modulational instability has also been executed which agree well with the
analytical results, thereby making the proposed system more feasible for
experimental realization.
