In this paper, we consider the modified one-dimensional Schrödinger equation:
$$(D_t-F(D))u=λ|u|^2u,$$
where F(ξ) is a second order constant coefficients classical elliptic symbol, and with 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 get a one term asymptotic expansion for $u$ when $t→+∞$.