(untitled) (Analysis in Theory and Applications, 2026)
We study the long time behavior of the modified one-dimensional deriva- tive Schrödinger equation (D_t - F(D))u = D_x(|u|^2u), where F(ξ) is a nonnegative second order constant coefficient elliptic symbol. For any smooth initial datum of size ε 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.