  The high-energy evolution in perturbative QCD suffers from a severe
lack-of-convergence problem, due to higher order corrections enhanced by double
and single transverse logarithms. We resum double logarithms to all orders
within the non-linear Balitsky-Kovchegov equation, by taking into account
successive soft gluon emissions strongly ordered in lifetime. We further resum
single logarithms generated by the first non-singular part of the splitting
functions and by the one-loop running of the coupling. The resummed BK equation
admits stable solutions, which are used to successfully fit the HERA data at
small $x$ for physically acceptable initial conditions and reasonable values of
the fit parameters.
