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We study the asymptotic behavior of the modified two-dimensionalSchr\"{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 asecond order constant coefficients elliptic symbol. For any smooth initialdatum of size $\varepsilon\ll1$, we prove that the solution is global-in-time,combining the vector fields method and a semiclassical analysis methodintroduced by Delort. Moreover, we present the pointwise decay estimates andthe large time asymptotic formulas of the solution.

Global existence and asymptotics for the modified two-dimensional Schr\"{o}dinger equation in the critical regime