  In this paper, we investigate for the first time, the spatiotemporal dynamics
of self-similar parabolic pulse evolution in multimode fibers (MMFs). We
present numerical predictions of the existence of parabolic pulse in a graded
and step index MMFs and demonstrate that the parabolic pulse formation process
could have a prominent effect on the spatiotemporal behavior in both fibers. We
study the effect of the input pulse energy and its initial modal composition on
the resulting parabolic pulse and the corresponding spatial beam profile. When
the fundamental mode is mostly excited, the pulse evolves into a linearly
chirped pulse with parabolic intensity shape that propagates self-similarly in
both fibers.This allows for efficient and high-quality pulse compression with
sub-40 fs pulse duration. Dependence on input pulse shapes, its chirp parameter
and sign towards conversion into parabolic shape are also reported. We quantify
the reshaping of the parabolic pulse using the misfit parameter. We demonstrate
a route to the parabolic pulse evolutions for all modes equal energy
distributed initial condition. In addition, we also observe that the beam
profiles of output fields could be different for same MMF with different
initial pulse energy. Moreover, a spatiotemporal nonlinear dynamic,
auto-selection beam of one specific mode is investigated. Thus remain
spatio-temporally stable for more than a decade of the input pulse energy
variation. Parabolic pulse formation process plays critical roles in these
nonlinear dynamics. This approach provides another framework to understand the
complex nonlinear dynamics in MMFs.
