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 L2(ℝ). We first prove that the standard fixed-profile modified-scattering ansatz fails at the unweighted L2 endpoint for the one-dimensional defocusing cubic nonlinear Schrödinger equation. More precisely, there exists a real-valued datum u*∈ L1(ℝ)∩⋂m∈ℕHm(ℝ), xu*∉ L2(ℝ)., for which the corrected Fourier profile has no strong L2 limit. Secondly, for every L2 datum, we prove generalized modified scattering: in the physical ray coordinates x=2tz, the rescaled solution admits a strong L2 asymptotic representation whose amplitude is the square root of the conserved transmission density and whose unit complex phase is determined by the initial datum.
Related
- Weighted Sobolev regularity and failure of profile convergence associated with the scattering operator for the three-dimensional cubic Klein-Gordon equation
- Global well-posedness and scattering for the defocusing nonlinear Schrödinger equation below energy space
- Global well-posedness and scattering for the three-dimensional defocusing cubic Schrödinger equation in Hs, s>1/2
- Global well-posedness and scattering of energy-critical defocusing nonlinear Schrödinger equation on high-dimensional hyperbolic spaces
- Large-data modified wave operators for the defocusing nonlinear Schrödinger equation in one space dimension with subcritical long-range nonlinearity
BibTeX
@misc{chen2026failure,
title = {{Failure of fixed-profile modified scattering and generalized modified scattering for the 1D defocusing cubic NLS}},
author = {Xi Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.02534},
eprint = {2608.02534},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02534},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Failure of fixed-profile modified scattering and generalized modified scattering for the 1D defocusing cubic NLS
AU - Xi Chen
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2608.02534
UR - https://arxiv.org/abs/2608.02534
AB - We study the long-time behavior of the one-dimensional defocusing cubic nonlinear Schrödinger equation with initial data in L^2(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_*∈ L^1(ℝ)∩⋂_m∈ℕH^m(ℝ), xu_*∉ L^2(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.
ER -