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In this article, we prove the global well-posedness and scattering of thecubic 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 groundstate, and derive the threshold of the global well-posedness and scattering.Then we show the global well-posedness and scattering below the threshold bythe concentration-compactness/rigidity method, where the almost periodicsolution is excluded by adapting the argument in the proof of the mass-criticalnonlinear Schr\"odinger equations by B. Dodson. As a byproduct of thescattering of the cubic focusing infinite coupled nonlinear Sch\"odingersystem, we obtain the scattering of the cubic focusing nonlinear Schr\"odingerequation on the small cylinder, this is the first large data scattering resultof the focusing nonlinear Schr\"odinger equations on the cylinders. In thearticle, we also show the global well-posedness and scattering of the twodimensional $N-$coupled focusing cubic nonlinear Schr\"odinger system in$\left(L^2(\mathbb{R}^2) \right)^N$.

Global well-posedness and scattering of the two dimensional cubic focusing nonlinear Schr\"odinger system