  An analytical study with respect to the nonlinear corrections for the nuclear
gluon distribution function in the next-to-leading order approximation at small
$x$ is presented. We consider the nonlinear corrections to the nuclear gluon
distribution functions at low values of $x$ and $Q^{2}$ using the
parametrization $F_{2}(x,Q^{2})$ and using the nuclear modification factors
where they have been obtained with the
Khanpour-Soleymaninia-Atashbar-Spiesberger-Guzey model. The CT18 gluon
distribution is used as baseline proton gluon density at
$Q_{0}^{2}=1.69~\mathrm{GeV}^2$. We discuss the behavior of the gluon densities
in the next-to-leading order and the next-to-next-to-leading order
approximations at the initial scale $Q_{0}^{2}$, as well as the modifications
due to the nonlinear corrections. We find the QCD nonlinear corrections are
significant for the next-to-leading order accuracy than the
next-to-next-to-leading order for light and heavy nuclei. The results of the
nonlinear GLR-MQ evolution equation are similar to those obtained with the
Rausch-Guzey-Klasen gluon upward and downward evolutions within the
uncertainties. The magnitude of the gluon distribution with the nonlinear
corrections increases with a decrease of $x$ and an increase of the atomic
number A.

High Energy Physics - Phenomenology