An approximate approach to the nonlinear DGLAP evaluation equation

We determined the effects of the first nonlinear corrections to the gluon distribution using the solution of the QCD nonlinear Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (NLDGLAP) evolution equation at small x. By using a Laplace-transform technique, the behavior of the gluon distribution is obtained by solving the Gribov, Levin, Ryskin, Mueller and Qiu (GLR-MQ) evolution equation with the nonlinear shadowing term incorporated. We show that the strong rise that is corresponding to the linear QCD evolution equations at small x can be tamed by screening effects. Consequently, the nonlinear effects for the gluon distributions are calculated and compared with the results for the integrated gluon density from the Balitsky-Kovchegov (BK) equation. The resulting analytic expression allows us to predict the shadowing correction to the logarithmic derivative $F_2(x,Q^2)$ with respect to $ln Q^{2}$ and to compare the results with H1 data and a QCD analysis fit.