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

Abstract

We study the long-time behavior of the one-dimensional defocusing cubic nonlinear Schrödinger 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 Schrödinger equation. More precisely, there exists a real-valued datum $$u_{\ast }\in L^1(\mathbb{R})\cap \bigcap _{m\in \mathbb{N}}{H}^m(\mathbb{R}),x{u}_{\ast }\notin L^2(\mathbb{R}).,$$ 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.

arXiv commentGeneralized modified scattering is provided in v2