Long Time Behavior of the Modified One-Dimensional Derivative Schrödinger Equation

We study the long time behavior of the modified one-dimensional derivative Schrödinger equation $(D_t - F(D))u = D_x(|u|^2 u)$, where $F(\xi)$ is a nonnegative second order constant coefficient elliptic symbol. For any smooth initial datum of size $\varepsilon \ll 1$, 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.