Asymptotics for the cubic 1D NLS with a slowly decaying potential

We consider the asymptotics of the one-dimensional cubic nonlinear Schr\"odinger equation with an external potential $V$ that does not admit bound states. Assuming that $\jBra{x}^{2+}V(x) \in L^1$ and that $u$ is orthogonal to any zero-energy resonances, we show that solutions exhibit modified scattering behavior. Our decay assumptions are weaker than those appearing in previous work and are likely nearly optimal for exceptional potentials, since the M\o{}ller wave operators are not known to be bounded on $L^p$ spaces unless $\jBra{x}^2V \in L^1$. Our method combines two techniques: First, we prove a new class of linear estimates, which allow us to relate vector fields associated with the equation with potential to vector fields associated with the flat equation, up to localized error terms. By using the improved local decay coming from the zero-energy assumption, we can treat these errors perturbatively, allowing us to treat the weighted estimates using the same vector field arguments as in the flat case. Second, we derive the asymptotic profile using the wave packet testing method introduced by Ifrim and Tataru. Again, by taking advantage of improved local decay near the origin, we are able to treat the potential as a lower-order perturbation, allowing us to use tools developed to study the flat problem.