Derivation of a Nonlinear Schr\"odinger Equation with a General power-type nonlinerity

In this paper we study the derivation of a certain type of NLS from many-body interactions of bosonic particles. We consider a model with a finite linear combination of $n$-body interactions, where $n \geq 2$ is an integer. We show that the $k$-particle marginal density of the BBGKY hierarchy converges when particle number goes to infinity, and the limit solves a corresponding infinite Gross-Pitaevskii hierarchy. We prove the uniqueness of factorized solution to the Gross-Pitaevskii hierarchy based on a priori space time estimates. The convergence is established by adapting the arguments originated or developed in \cite{ESY}, \cite{KSS} and \cite{CPquintic}. For the uniqueness part, we expand the procedure followed in \cite{KM} by introducing a different board game argument to handle the new contraction operator. This new board game argument helps us obtain a good estimate on the Duhamel terms. In \cite{KM}, the relevant space time estimates are assumed to be true, while we give a prove for it.