New exact analytical solution of the nonlinear Gribov-Levin-Ryskin-Mueller-Qiu equation

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 $Q_s^2\,\propto\,Q_0^2 e^{\lambda Y}$.