Global well-posedness and scattering of the two dimensional cubic focusing nonlinear Schrödinger system

In this article, we prove the global well-posedness (GWP) and scattering of the cubic focusing infinitely coupled nonlinear Schr & ouml;dinger system (NLSS) on R2 below the ground state in L2xh1(R2 x Z). We first establish the variational characterization of the ground state and derive the threshold for global well-posedness and scattering. We then demonstrate the global well-posedness and scattering below the threshold using the concentration-compactness/rigidity method. The almost periodic solution is excluded by adapting the argument used in the proof of the focusing mass-critical nonlinear Schr & ouml;dinger equations (NLS) by B. Dodson. As a byproduct of the scattering of the cubic focusing infinitely coupled nonlinear Schr & ouml;dinger system, we obtain the scattering of the cubic focusing nonlinear Schr & ouml;dinger equation on the small cylinder. We also show the global well-posedness and scattering of the two dimensional N-coupled focusing cubic nonlinear Schr & ouml;dinger system in (L2(R2))N. (c) 2025 Published by Elsevier Inc.