Nonlinear corrections on the parametrization methods

We present nonlinear corrections (NLCs) to the distribution functions at low values of x and $$Q^{2}$$ Q 2 using the parametrization $$F_{2}(x,Q^{2})$$ F 2 ( x , Q 2 ) and $$F_{L}(x,Q^{2})$$ F L ( x , Q 2 ) . We use a direct method to extract nonlinear corrections to the ratio of structure functions and the reduced cross section in the next-to-next-to-leading order (NNLO) approximation with respect to the parametrization method (PM). Comparisons between the nonlinear results with the bounds in the color dipole model (CDM) and HERA data indicate the consistency of the nonlinear behavior of the gluon distribution function at low x and low $$Q^{2}$$ Q 2 . The nonlinear longitudinal structure functions are comparable with the H1 Collaboration data in a wide range of $$Q^{2}$$ Q 2 values. Consequently, the nonlinear corrections at NNLO approximation to the reduced cross sections at low and moderate $$Q^{2}$$ Q 2 values show good agreement with the HERA combined data. These results at low x and low $$Q^{2}$$ Q 2 can be applied to the LHeC region for analyses of ultra-high-energy processes.