site in each of the two systems. We evolve one or the other systems by steps of one unit in $y$. We measure the position of the traveling wave front by searching, at each rapidity $y$, the rightmost site $i_F$ for which $S(y,i_F)<0.5$. We then define the front position ${\cal I}_y$ from a linear interpolation between $i_F$ and $i_F+1$.

Let us move on to the results.

$S$-matrix
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We compute the $S$-matrix in 3 different variants of the model: bare multiple scatterings in the lab frame (which is the BK assumption; $\sigma=0$), the same but in the center-of-mass frame (which is Mueller’s unitarization procedure; $\sigma=\frac12$) and multiple scatterings off boost-invariant saturated wave functions in the center-of-mass frame ($S$ should be boost invariant in this case: We will check it later).

Tuning the frame in the Monte-Carlo is technically easy: It is enough to share the rapidity evolution of the projectile and of the target proportionally to $\sigma$.

The results are presented in Fig. \[fig:s\_20\]. We see that rapidity evolution starts with the blackening of the central region around $i\sim 0$. At $y\sim 15-20$ two traveling waves form, and propagate symmetrically towards larger (resp. smaller) values of $i$. In the following, 