Failure of fixed-profile modified scattering and generalized modified scattering for the 1D defocusing cubic NLS

We study the long-time behavior of the one-dimensional defocusing cubic nonlinear Schr"odinger equation with initial data in $L^2(\mathbb{R})$. We first prove that the standard fixed-profile modified-scattering ansatz fails at the unweighted $L^2$ endpoint for the one-dimensional defocusing cubic nonlinear Schr"odinger equation. More precisely, there exists a real-valued datum $$u_{\ast }\in L^1(\mathbb{R})\cap \bigcap _{m\in \mathbb{N}}{H}^m(\mathbb{R}),x{u}_{\ast }\mathrm{\notin }{L}^2(\mathbb{R})\mathrm{.},$$ for which the corrected Fourier profile has no strong $L^2$ limit. Secondly, for every $L^2$ datum, we prove generalized modified scattering: in the physical ray coordinates $x=2tz$, the rescaled solution admits a strong $L^2$ asymptotic representation whose amplitude is the square root of the conserved transmission density and whose unit complex phase is determined by the initial datum.