  In this paper, the proton structure function $F_2 ^p (x,Q^2)$ at small-$x$ is
investigated using an analytical solution of the Balitsky-Kovchegov (BK)
equation. In the context of the color dipole description of deep inelastic
scattering (DIS), the structure function $F_2 ^p (x,Q^2)$ is computed by
applying the analytical expression for the scattering amplitude $N(k,Y)$
derived from the BK solution. At transverse momentum $k$ and total rapidity
$Y$, the scattering amplitude $N(k,Y)$ represents the propagation of the
quark-antiquark dipole in the color dipole description of DIS. Using the BK
solution we extracted the integrated gluon density $xg(x,Q^2)$ and then
compared our theoretical estimation with the LHAPDF global data fits NNPDF3.1sx
and CT18. Finally, we investigated the behavior of $F_2 ^p (x,Q^2)$ in the
kinematic region of $10^{-5} \leq x \leq 10^{-2}$ and $2.5$ $GeV^{2}$ $\leq$
$Q^2$ $\leq$ $60$ $GeV^{2}$. Our predicted results for $F_2 ^p (x,Q^2)$ within
the specified kinematic region are in good agreement with the recent
high-precision data for $F_2 ^p (x,Q^2)$ from HERA (H1 Collaboration) and the
LHAPDF global parametrization group NNPDF3.1sx.
