nucleus. We assume that the photon splits into a quark-antiquark pair, on which we apply the dipole formulation, that is we work in the large $N_c$ and the leading logarithmic limits. This means that the photon will develop a cascade of color dipoles, which interact with the nucleus. The nucleus is assumed to be at rest. We assume that each dipole interacts with the nucleus by a two gluon exchange, independent from the rest of the dipoles. The nucleus is approximated by a cylinder which makes the calculations easier. The multiple pomeron exchanges will correspond to the simultaneous interactions of multiple dipoles with the nucleus, which means that the $k$:th pomeron exchange corresponds to $k$ dipoles simultaneously interacting with the nucleus by exchanging $k$ color neutral gluon pairs. With these considerations and summing pomeron exchanges up to all orders, Kovchegov, in [@e6], derived the following non-linear differential-integral equation for the forward scattering amplitude off the nucleus: $$\begin{aligned}
\frac{\partial N(\bold{x}_{10},\bold{b}_0,Y)}{\partial
  Y}&=&\frac{\bar{\alpha}}{2\pi}\int_\rho
  d^2\bold{x}_2\frac{x_{10}^2}{x_{20}^2x_{12}^2}\biggl(
  N(\bold{x}_{20},\bold{b}_0 +
  \frac{1}{2}\bold{x}_{12},Y) + \label{eq:BK}\nonumber \\ 
& & +N(\bold{x}_{12},\bold{b}_0 -
  \frac{1}{2}\bold{x}_{20},Y)-N(\bold{x}_{10},\bold{b}_0,Y)- \\
& & -N(\bold{x}_{20},\bold{b}_0 +
  \frac{1}{2}\bold{x}_{12},Y)N(\bold{x}_{12},\bold{b}_0 -
  \frac{1}{2}\bold{x}_{20},Y)\biggr)\nonumber \end{aligned}$$

Here the imaginary part of $N$ gives the total cross section. This equation is known as the Balistky-Kovchegov (BK) 