Dynamics of large boson systems with attractive interaction and a derivation of the cubic focusing NLS equation in R3

We consider a system of N bosons where the particles experience a short-range two-body interaction given by N−1vN(x) = N3β−1v(Nβx), where v∈Cc∞(R3), without a definite sign on v. We extend the results of Grillakis and Machedon [Commun. Math. Phys. 324(2), 601–636 (2013)] and Kuz [Differ. Integral Equations 30(7/8), 587–630 (2017)] regarding the second-order correction to the mean-field evolution of systems with repulsive interaction to systems with attractive interaction for 0<β<12. Our extension allows for a more general set of initial data, which includes coherent states. We also provide both a derivation of the focusing nonlinear Schrödinger equation in 3D from the many-body system and its rate of convergence toward mean-field for 0<β<13.