Decay estimates for Schrödinger systems with time-dependent potentials in 2D

We consider the Cauchy problem for systems of nonlinear Schrödinger equations with time-dependent potentials in 2D. Under assumptions about mass resonances and potentials, we prove the global existence of the nonlinear Schrödinger systems with small initial data. In particular, by analyzing the operator $\mathrm{\Delta }$ and time-dependent potentials $V_j$ separately, we show that the small global solutions satisfy time decay estimates of order $O((t\log t{)}^{-1})$ when $p=2$, and the small global solutions satisfy time decay estimates of order $O(t^{-1})$ when $p>2$.