Nonlinear correction to the longitudinal structure function at small x

We computed the longitudinal proton structure function $F_{L}$, using the nonlinear Dokshitzer-Gribov-Lipatov-Altarelli-parisi (NLDGLAP) evolution equation approach at small $x$. For the gluon distribution, the nonlinear effects are related to the longitudinal structure function. As, the very small $x$ 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, the strong rise that is corresponding to the linear QCD evolution equations, can be tamed by screening effects. Consequently, the obtained longitudinal structure function shows a tamed growth at small $x$. We computed the predictions for all detail of the nonlinear longitudinal structure function in the kinematic range where it has been measured by $H1$ collaboration and compared with computation Moch, Vermaseren and Vogt at the second order with input data from MRST QCD fit.