  We investigate in detail the qualitative similarities between the pulse
localization characteristics observed using sinusoidal phase modulation during
linear propagation and those seen during the evolution of Akhmediev breathers
during propagation in a system governed by the nonlinear Schr{\"o}dinger
equation. The profiles obtained at the point of maximum focusing indeed present
very close temporal and spectral features. If the respective linear and
nonlinear longitudinal evolutions of those profiles are similar in the vicinity
of the point of maximum focusing, they may diverge significantly for longer
propagation distance. Our analysis and numerical simulations are confirmed by
experiments performed in optical fiber.
