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

Abstract

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 classical elliptic symbol. For any smooth initial datum of size \(\varepsilon\ll1\), we prove that the solution is still global-in-time when the problem does not admit the classical vector fields. Moreover, we present the pointwise decay estimates and the large time asymptotic formulas of the solution.