Scattering for the $1d$ NLS with inhomogeneous nonlinearities

Abstract

We prove large-data scattering in $H^1$ for inhomogeneous nonlinear Schr\"odinger equations in one space dimension for powers $p>2$. We assume the inhomogeneity is nonnegative and repulsive; we additionally require decay at infinity in the case $2<p\leq 4$. We use the method of concentration-compactness and contradiction, utilizing a Morawetz estimate in the style of Nakanishi in order to preclude the existence of compact solutions.