In this paper, we study the derivation of a certain type of NLS from many-body interactions of bosonic particles in $d=1,2$. We consider a model with a finite linear combination of $n$-body interactions and obtain that the $k$-particle marginal density of the BBGKY hierarchy converges when particle number goes to infinity. Moreover, 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 ,  and . For the uniqueness part, we expand the procedure in  by introducing a different board game argument to handle the factorial in the number of terms from Duhamel expansion. The space time bound assumption in  is removed in our proof.
Zhihui Xie. "Derivation of a nonlinear Schrödinger equation with a general power-type nonlinearity in $d=1,2$." Differential Integral Equations 28 (5/6) 455 - 504, May/June 2015. https://doi.org/10.57262/die/1427744097