[@MT] for the pioneering calculations, but without the connection to traveling waves) or to reviews in mathematical physics [@VS] for the details. Traveling waves have a phenomenological signature in high-energy scattering, called “geometric scaling” and found in deep-inelastic scattering data [@SGBK]. We give here the technical features of our particular model without much justification.

When $i$ and $y$ are large, $$S_\text{BK}(y,i)\sim 1-e^{-\gamma_0(i-{\cal I}_y)},
\label{formSBK}$$ where ${\cal I}_y$ is the position of the wave front at rapidity $y$. In the context of QCD, ${\cal I}_y$ would be the logarithm of the squared saturation scale.

As well-known, $\gamma_0$ and ${\cal I}_y$ are determined from an analysis of the linearized equation (\[generating\]). We insert Eq. (\[formSBK\]) into Eq. (\[generating\]), and after linearization and some easy algebra, we arrive at the expression for the velocity $V$ of a front of the form (\[formSBK\]) with decay rate $\gamma$: $$V={\cal I}_{y+1}-{\cal I}_y=\frac{\chi(\gamma)}{\gamma},$$ where $$\begin{gathered}
\chi(\gamma)=
\ln\bigg[
\frac12+\frac12\left(1-\frac{1}{e}\right)\times\\
\times\left(\frac{1}{1-e^{\gamma-1}}+\frac{1}{1-e^{-\gamma-1}}\right)
\bigg]\end{gathered}$$ is the characteristic function of the evolution kernel in Eq. (\[generating\]). The relevant value of $\gamma$ at large rapidity is the one that minimizes $V(\gamma)$. This minimization gives $$\gamma_0=0.607187\cdots\ ,\ \
V_0=1.02935\cdots$$ which are the decay rate and the velocity of the front at infinite rapidity. The 