$$y_F\simeq 7.4,\ y_R\simeq 2.2\ \ \text{for}\ 
1/\alpha^2=20.$$ We see in Fig. \[fig:bk\] that first the velocity is 0, which corresponds to the phase in which the parton numbers are building up through a linear evolution. (The linear phase is named in QCD after Balitsky, Fadin, Kuraev and Lipatov (BFKL) [@BFKL]). Later, after about 10 units of rapidity (which is the order of magnitude of $y_F$), a sharp peak appears and decays over a few units of $y$. This peak may be interpreted as follows: In the initial stage of the evolution, when the front is building up, its shape is steeper than the asymptotic shape (\[formSBK\]) in the region where the position is measured ($S\sim 0.5$), and hence its velocity is larger than $V_0$. Then, the front relaxes to its asymptotic shape in that region, which takes of the order of $y_R$ steps of rapidity. In the final stage, for $y\gg 10$, the asymptotic velocity is approached in an algebraic way, in good agreement with the theoretical expectations of Eq. (\[Vy\]).

Front velocity in different models
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We want to compare bare unitarization through multiple scatterings in different frames to the variant of the model in which saturation is included 