in a boost-invariant way. The front velocity for different schemes and $1/\alpha^2$ set to 20 is shown in Fig. \[fig:v20\_1\].

Bare multiple scatterings lead to a dip around $y\sim 20$ whose depth is maximum for $\sigma=\frac12$. At large $y$, the velocity $V_0$ is reached algebraically, whatever the frame is. At any finite $y$, this model gives rise to different front solutions in different frames: Boost invariance is manifestly badly broken.

Solutions with saturation in the evolution look like expected: The velocity rapidly reaches a plateau, with a value that is lower than that of the asymptotic BK velocity. Theoretically, this should happen after typically $y_F+y_R$ units of rapidity. For low values of $1/\alpha^2$, the formula giving $y_R$ is expected to need large corrections, and so the numerical value of $y_R$ is not trustable. The value of the asymptotic velocity is equal in all frames, and except for the lab frame, all curves superimpose reasonably well during the whole rapidity evolution. We deduce that boost invariance is quite well preserved with our saturation solution, although not perfectly. We attribute the lack of complete superposition of the curves to the explicit breaking of boost-invariance due to our discretization in rapidity.

The same 