Testing Non-linear Evolution with Running Coupling Corrections in $ep$ and $pp$ collisions

The perturbative QCD predicts that the growth of the gluon density at small-$x$ (high energies) should saturate, forming a Color Glass Condensate (CGC), which is described in mean field approximation by the Balitsky-Kovchegov (BK) equation. Recently, the next-to-leading order corrections for the BK equation were derived and a global fit of the inclusive $ep$ HERA data was performed, resulting in a parameterization for the forward scattering amplitude. In this paper we compare this parameterization with the predictions of other phenomenological models and investigate the saturation physics in diffractive deep inelastic electron-proton scattering and in the forward hadron production in $pp$ collisions. Our results demonstrate that the running coupling BK solution is able to describe these observables.