  The most spiral galaxies have a flat rotational velocity curve, according to
the different observational techniques used in several wavelengths domain. In
this work, we show that non-linear terms are able to balance the dispersive
effect of the wave, thus reviving the observed rotational curve profiles
without inclusion of any other but baryonic matter concentrated in the bulge
and disk. In order to prove that the considered model is able to restore a flat
rotational curve, Milky Way has been chosen as the best mapped galaxy to apply
on. Using the gravitational N-body simulations with up to $10^7$ particles, we
test this dynamical model in the case of the Milky Way with two different
approaches. Within the direct approach, as an input condition in the simulation
runs we set the spiral surface density distribution which is previously
obtained as an explicit solution to non-linear Schr\"{o}dinger equation
(instead of a widely used exponential disk approximation). In the evolutionary
approach, we initialize the runs with different initial mass and rotational
velocity distributions, in order to capture the natural formation of spiral
arms, and to determine their role in the disk evolution. In both cases we are
able to reproduce the stable and non-expanding disk structures at the
simulation end times of $\sim10^9$ years, with no halo inclusion. Although the
given model doesn't take into account the velocity dispersion of stars and
finite disk thickness, the results presented here still imply that non-linear
effects can significantly alter the amount of dark matter which is required to
keep the galactic disk in stable configuration.
