  The higher-order perturbative corrections, beyond leading logarithmic
accuracy, to the BFKL evolution in QCD at high energy are well known to suffer
from a severe lack-of-convergence problem, due to radiative corrections
enhanced by double collinear logarithms. Via an explicit calculation of Feynman
graphs in light cone (time-ordered) perturbation theory, we show that the
corrections enhanced by double logarithms (either energy-collinear, or double
collinear) are associated with soft gluon emissions which are strictly ordered
in lifetime. These corrections can be resummed to all orders by solving an
evolution equation which is non-local in rapidity. This equation can be
equivalently rewritten in local form, but with modified kernel and initial
conditions, which resum double collinear logs to all orders. We extend this
resummation to the next-to-leading order BFKL and BK equations. The first
numerical studies of the collinearly-improved BK equation demonstrate the
essential role of the resummation in both stabilizing and slowing down the
evolution.
