Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $N^{-1}v_N(x)=N^{3\beta-1}v(N^\beta x)$
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
$0<\beta<\frac{1}{2}$. 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. Napi\'orkowski,
Adv. Theor. Math. Phys., $\textbf{21}$, 683(2017), we also provide both a
derivation of the focusing nonlinear Schr\"odinger equation (NLS) in $3$D from
the many-body system and its rate of convergence toward mean-field for
$0<\beta<\frac{1}{3}$. In particular, we give two derivations of the focusing
NLS, one based on the $N$-norm approximation given in the work of Nam and
Napi\'orkowski 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 $0<\beta<\frac{1}{6}$.

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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: cs
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: quant-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: q-bio
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ph