  We construct spatiotemporal localized envelope solutions of a
(3+1)-dimensional nonlinear Schr\"{o}dinger equation with varying coefficients
such as dispersion, nonlinearity and gain parameters through similarity
transformation technique. The obtained localized rational solutions can serve
as prototypes of rogue waves in different branches of science. We investigate
the characteristics of constructed localized solutions in detail when it
propagates through six different dispersion profiles, namely constant, linear,
Gaussian, hyperbolic, logarithm and exponential. We also obtain expressions for
the hump and valleys of rogue wave intensity profiles for these six dispersion
profiles and study the trajectory of it in each case. Further, we analyze how
the intensity of another localized solution, namely breather, changes when it
propagates through the aforementioned six dispersion profiles. Our studies
reveal that these localized solutions co-exist with the collapsing solutions
which are already found in the (3+1)-dimensional nonlinear Schr\"{o}dinger
equation. The obtained results will help to understand the corresponding
localized wave phenomena in related fields.
