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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.

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We show that the nonlinear corrections to the longitudinal structure function can be tamed the singularity behavior at low x values, with respect to GLR-MQ equations. This approach can determined the shadowing longitudinal structure function based on the shadowing corrections to the gluon and singlet quark structure functions. Comparing our results with HERA data show that at very low x this behavior completely tamed by these corrections.

The behavior of the charm and bottom structure functions (F i k (x, Q 2 ), i=c,b; k=2,L) at small-x is considered with respect to the hard-Pomeron and saturation models. Having checked that this behavior predicate the heavy flavor reduced cross sections concerning the unshadowed and shadowed corrections. We will show that the effective exponents for the unshadowed and saturation corrections are independent of x and Q 2 , and also the effective coefficients are dependent to ln Q 2 compared to Donnachie-Landshoff (DL) and color dipole (CD) models.

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We present a set of formulae to extract two second-order independent differential equations for the gluon and singlet distribution functions. Our results extend from the LO up to NNLO DGLAP evolution equations with respect to the hard-Pomeron behavior at low-x. In this approach, both singlet quarks and gluons have the same high-energy behavior at low-x. We solve the independent DGLAP evolution equations for the functions F INTRODUCTIONThe Dokshitzer-Gribov-Lipatov-AltarelliParisi (DGLAP) [1] evolution equations are fundamental tools to study the Q 2 -and xevolutions of structure functions, where x is the Bjorken scaling parameter and Q 2 is the virtuality of the exchanged vector boson in a deep inelastic scattering process [2]. The measurements of the F 2 (x, Q 2 ) structure functions by deep inelastic scattering processes in the low-x region have opened a new era in parton density measurements inside hadrons. The structure function reflects the momentum distributions of the partons in the nucleon. It is also important to know the gluon distribution inside a hadron at low-x because gluons are expected to be dominant in this region. The steep increase of F 2 (x, Q 2 ) towards low-x observed at the hadron electron ring accelerator (HERA) also indicates a similar increase in the gluon distribution towards low-x in perturbative quantum chromodynamics. In the usual procedure, the deep inelastic scattering data are analyzed by the next-to-next-to-leading order (NNLO) QCD fits based on the numerical solution of the DGLAP evolution equations, and it has been found that * Electronic address: grboroun@gmail.com; boroun@razi.ac.ir † brezaei@razi.ac.ir the DGLAP analysis can well describe the data in the perturbative region Q 2 ≥ 1GeV 2 [3]. As an alternative to the numerical solution, one can study the behavior of quarks and gluons via analytic solutions of the evolution equations. Although exact analytic solutions of the DGLAP equations cannot be obtained in the entire range of x and Q 2 , such solutions are possible under certain conditions and are quite successful as far as the HERA low-x data are concerned. Some of these methods [4] were proposed in the literature by using expansion method or pomeron behavior. The low-x region of DIS offers a unique possibility to explore the Regge limit of pQCD [5]. This theory is successfully described by the exchange of a particle with appropriate quantum numbers and the exchanged particle is called a Regge pole. Phenomenologically, the Regge pole approach to DIS implies that the structure functions are sums of powers in x, modulus logarithmic terms, each with a Q 2 -dependent residue factor. Also, in the DGLAP formalism the gluon splitting functions are singular as x → 0. Thus, the gluon distribution will become large as x → 0, and its contribution to the evolution of the parton distribution becomes dominant. In particular, the gluon will drive the quark singlet distribution, and, hence, the structure function F 2 becomes large as well, the rise increasing in steepness as Q 2 incr...

An analytical solution of the QCD evolution equations for the singlet and gluon distribution is presented. We decouple DGLAP evolution equations into the initial conditions by using a Laplace transform method at N n LO analysis. The relationship between the nonlinear behavior and color dipole model is considered based on an effective exponent behavior at low-x values. We obtain the effective exponent at NLO analysis from the decoupled behavior of the distribution functions. The proton structure function compared with H1 data from the inclusive structure function F2(x, Q 2 ) for x≤ 10 −2 and 5≤Q 2 ≤250 GeV 2 . * Electronic address: brezaei@razi.ac.ir † Electronic address: