  We study the solution of the nonlinear BK evolution equation with the
recently calculated running coupling corrections [hep-ph/0609105,
hep-ph/0609090]. Performing a numerical solution we confirm the earlier result
of [hep-ph/0408216] that the high energy evolution with the running coupling
leads to a universal scaling behavior for the dipole scattering amplitude. The
running coupling corrections calculated recently significantly change the shape
of the scaling function as compared to the fixed coupling case leading to a
considerable increase in the anomalous dimension and to a slow-down of the
evolution with rapidity. The difference between the two recent calculations is
due to an extra contribution to the evolution kernel, referred to as the
subtraction term, which arises when running coupling corrections are included.
These subtraction terms were neglected in both recent calculations. We evaluate
numerically the subtraction terms for both calculations, and demonstrate that
when the subtraction terms are added back to the evolution kernels obtained in
the two works the resulting dipole amplitudes agree with each other! We then
use the complete running coupling kernel including the subtraction term to find
the numerical solution of the resulting full non-linear evolution equation with
the running coupling corrections. Again the scaling regime is recovered at very
large rapidity.
