  We analytically solve the Sudakov suppressed Balitsky-Kovchegov evolution
equation with the fixed and running coupling constants in the saturation
region. The analytic solution of the $S$-matrix shows the
$\exp(\mathcal{O}(\eta^2))$ rapidity dependence of the solution with the fixed
coupling constant is replaced by $\exp(\mathcal{O}(\eta^{3/2}))$ dependence in
the smallest dipole running coupling case rather than obeying the law found in
our previous publication, in which all the solutions of the next-to-leading
order evolution equations comply with $\exp(\mathcal{O}(\eta))$ rapidity
dependence once the QCD coupling is switched from the fixed coupling to the
smallest dipole running coupling prescription. This finding indicates that the
corrections of the sub-leading double logarithms in the Sudakov suppressed
evolution equation are significant, which compensate part of the evolution
decrease of the dipole amplitude made by running coupling effect. To test the
analytic findings, we calculate the numerical solutions of the Sudakov
suppressed evolution equation, the numerical results confirm the analytic
outcomes. Moreover, we use the numerical solutions of the evolution equation to
fit the HERA data. It shows that the Sudakove suppressed evolution equation can
give good quality fit to the data.
