  We present results from analytic solutions to the running coupling, full
next-to-leading order, and collinearly improved next-to-leading order
Balitsky-Kovchegov equations in the saturation region with the smallest dipole
size QCD running coupling prescription. The analytic results of the $S$-matrix
of the latter two equations show that the $\exp(-\mathcal{O}(Y^{3/2}))$
rapidity dependence of the solutions are replaced by $\exp(-\mathcal{O}(Y))$
dependence once the running coupling prescription is switched from parent
dipole to the smallest dipole prescription, which indicate that the $S$-matrix
has a strong dependence on the choice of running coupling prescription. We
compute the numerical solutions of these Balitsky-Kovchegov equations with the
smallest and parent dipole running coupling prescriptions, the numerical
results confirm the analytic outcomes. The rare fluctuations of the $S$-matrix
on top of next-to-leading order corrections are also studied under the smallest
dipole running coupling prescription in the center of mass frame. It shows that
the rare fluctuations are strongly suppressed and less important in the
smallest dipole running coupling prescription case as compared to the parent
dipole running coupling prescription case.
