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Dynamics of Large Boson Systems with Attractive Interaction and a Derivation of the Cubic Focusing NLS in $\mathbb{R}^3$

We consider a system of $N$ bosons where the particles experience a short range two-body interaction given by $N^{-1}v_N(x)=N^{3β-1}v(N^βx)$ where $v \in C^\infty_c(\mathbb{R}^3)$, without a definite sign on $v$. We extend the results of M. Grillakis and M. Machedon, Comm. Math. Phys., $\textbf{324}$, 601(2013) and E. Kuz, Differ. Integral Equ., $\textbf{137}$, 1613(2015) regarding the second-order correction to the mean-field evolution of systems with repulsive interaction to systems with attractive interaction for $0<β<\frac{1}{2}$. Our extension allows for a more general set of initial data which includes coherent states. Inspired by the works of X. Chen and J. Holmer, Arch. Ration. Mech. Anal., $\textbf{221}$, 631(2016) and Int. Math. Res. Not., $\textbf{2017}$, 4173(2017), and P. T. Nam and M. Napiórkowski, Adv. Theor. Math. Phys., $\textbf{21}$, 683(2017), we also provide both a derivation of the focusing nonlinear Schrödinger equation (NLS) in $3$D from the many-body system and its rate of convergence toward mean-field for $0<β<\frac{1}{3}$. In particular, we give two derivations of the focusing NLS, one based on the $N$-norm approximation given in the work of Nam and Napiórkowski and the other via a method introduced in P. Pickl, J. Stat. Phys., $\textbf{140}$, 76(2010). The techniques used in this article are standard in the literature of dispersive PDEs. Nevertheless, the derivation of the focusing NLS had only previously been studied for the 1D \& 2D cases and conditionally answered for the 3D case for $0<β<\frac{1}{6}$.