Classification of the minimal-mass blowup solutions to the two dimensional focusing cubic nonlinear Schr\"odinger system

In this article, we study the two dimensional focusing finitely and infinitely coupled cubic nonlinear Schr"odinger system when the mass is equal to the scattering threshold. For the focusing finitely coupled cubic nonlinear Schr"odinger system, we present a complete classification of minimal-mass blowup solutions. Specifically, we demonstrate that all such solutions must be either solitons or their pseudo-conformal transformations. To prove this result, we develop a modulation analysis that accounts for multi-component interactions to overcome the multiply phase transformations caused by the multi-component. A long time Strichartz estimate for vector-valued solutions is established to solve the difficulty posed by the Galilean transformations and spatial translation, where a new vector-valued bilinear estimate is proven to address the challenges caused by the coupled nonlinear interaction. For the infinitely coupled focusing nonlinear Schr"odinger system when the mass is equal or slightly above the scattering threshold in \cite{CGHY}, we show that scattering is the only dynamical behavior of the solutions to the infinitely coupled system.