Pair excitations and the mean field approximation of interacting Bosons,   I
  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. 
M. Grillakis, M. Machedon
math-ph math.AP math.MP