  In this article, we prove the global well-posedness and scattering of the
cubic focusing infinite coupled nonlinear Schr\"odinger system on
$\mathbb{R}^2$ below the threshold in $L_x^2h^1(\mathbb{R}^2\times
\mathbb{Z})$. We first establish the variational characterization of the ground
state, and derive the threshold of the global well-posedness and scattering.
Then we show the global well-posedness and scattering below the threshold by
the concentration-compactness/rigidity method, where the almost periodic
solution is excluded by adapting the argument in the proof of the mass-critical
nonlinear Schr\"odinger equations by B. Dodson. As a byproduct of the
scattering of the cubic focusing infinite coupled nonlinear Sch\"odinger
system, we obtain the scattering of the cubic focusing nonlinear Schr\"odinger
equation on the small cylinder, this is the first large data scattering result
of the focusing nonlinear Schr\"odinger equations on the cylinders. In the
article, we also show the global well-posedness and scattering of the two
dimensional $N-$coupled focusing cubic nonlinear Schr\"odinger system in
$\left(L^2(\mathbb{R}^2) \right)^N$.
