  The GLR-MQ equation is a nonlinear evolution equation that takes into account
the shadowing effect, which tames the growth of the gluon at small-$x$. In this
study, we analytically solve for the first time the nonlinear GLR-MQ equation
using the homogeneous balance method. The definite solution of the GLR-MQ
equation is obtained by fitting the MSTW2008LO gluon distribution data. We find
that the geometric scaling is an intrinsic property of our analytical solution
and the gluon distribution functions from our solution are able to reproduce
the MSTW2008LO data. These results indicate that our analytical solution from
the homogeneous balance method is valid to describe the gluon behavior at
small-$x$. Moreover, the saturation scale $Q_s$ has been extracted from our
analytical solution, we find that the energy-dependent saturation scale obeys
the exponential law $Q_s^2\,\propto\,Q_0^2 e^{\lambda Y}$.
