Non-linear corrections to the longitudinal structure function $F_{L}$ from the parametrization of $F_{2}$: Laplace transform approach

The non-linear corrections (NLC) to the longitudinal structure function in a limited approach is derived at low values of the Bjorken variable $x$ by using the Laplace transforms technique. The non-linear behavior of the longitudinal structure function is determined with respect to the Gribov-Levin-Ryskin Mueller-Qiu (GLR-MQ) and Altarelli-Martinelli (AM) equations. These results show that the non-linear longitudinal structure function can be determined directly in terms of the parametrization of $F_{2}(x,Q^{2})$ and the derivative of the proton structure function with respect to $\ln{Q^{2}}$. These corrections improve the behavior of the longitudinal structure function at low values of $Q^{2}$ in comparison with other parametrization methods.