  In our previous work \cite{GMM1},\cite{GMM2} we introduced a correction to
the mean field approximation of interacting Bosons. This correction describes
the evolution of pairs of particles that leave the condensate and subsequently
evolve on a background formed by the condensate. In \cite{GMM2} we carried out
the analysis assuming that the interactions are independent of the number of
particles $N$. Here we consider the case of stronger interactions. We offer a
new transparent derivation for the evolution of pair excitations. Indeed, we
obtain a pair of linear equations describing their evolution. Furthermore, we
obtain apriory estimates independent of the number of particles and use these
to compare the exact with the approximate dynamics.
