  We study the asymptotic behavior of the modified two-dimensional
Schr\"{o}dinger equation $ (D_t -F(D))u=\lambda|u| u$ in the critical regime,
where $\lambda \in \mathbb{C}$ with $\text{Im} \lambda \ge0$ and $F(\xi)$ 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.
