Global existence and asymptotics for the modified two-dimensional Schrödinger equation in the critical regime

We study the asymptotic behavior of the modified two-dimensional Schrödinger equation $ (D_t -F(D))u=λ|u| u$ in the critical regime, where $λ\in \mathbb{C}$ with $\text{Im} λ\ge0$ and $F(ξ)$ is a second order constant coefficients elliptic symbol. For any smooth initial datum of size $\varepsilon\ll1$, we prove that the solution is global-in-time, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we present the pointwise decay estimates and the large time asymptotic formulas of the solution.