  We study a variable-coefficient nonlinear Schr\"odinger (vc-NLS) equation
with higher-order effects. We show that the breather solution can be converted
into four types of nonlinear waves on constant backgrounds including the
multi-peak solitons, antidark soliton, periodic wave and W-shaped soliton. The
transition condition requiring the group velocity dispersion (GVD) and
third-order dispersion (TOD) to scale linearly is obtained analytically. We
display several kinds of elastic interactions between the transformed nonlinear
waves. We discuss the dispersion management of multi-peak soliton, which
indicates that the GVD coefficient controls the number of peaks of the wave
while the TOD coefficient has compression effect. The gain or loss has
influence on the amplitudes of the multi-peak soliton. We further derive the
breather multiple births by using multiple compression points of Akhmediev
breathers in optical fiber systems with periodic dispersion. The number of ABs
depends on the amplitude of the modulation but not on its wavelength, which
affects their separation distance. In the limiting case, the breather multiple
births reduce to the Peregrine combs. We discuss the effects of TOD coefficient
on the spatiotemporal characteristics of Peregrine combs. When the amplitude of
the modulation is equal to 1, the Peregrine comb is converted into a Peregrine
wall that can be seen as intermediate state between rogue wave and W-shaped
soliton. We finally find that the modulational stability regions with zero
growth rate coincide with the transition condition using rogue wave
eigenvalues. Our results could be useful for the experimental control and
manipulation of the formation of generalized Peregrine rogue waves in diverse
physical systems modeled by vc-NLS equation with higher-order effects.
