Global flows for the cubic nonlinear Schrödinger equation without decay assumptions
Abstract
For each positive integer N, we construct global spectral group actions for the defocusing and focusing nonlinear Schrödinger hierarchies on initial-data classes containing W2N+1,∞(ℝ). For N≥2, the quadratic flow gives global classical cubic NLS solutions. The construction extends the Sato--Segal--Wilson and Kotani frameworks to matrix Dirac systems; a homogeneous tau-function identity supplies the focusing positivity. No decay, smallness, periodicity, arithmetic condition, or prescribed spectral background is required for the initial-data inclusion. As a principal application, we prove global well-posedness on M∞,15(ℝ) for both signs, with finite-time bounds and locally Lipschitz dependence on initial-data norm balls. The passage to this Banach-space flow uses near-real-axis Weyl asymptotics and continuity of normalized Toeplitz operators on compact initial-data sets to obtain uniform spatial bounds from five bounded derivatives. In the defocusing case, the flow extends to M∞,15(ℝ)+H1(ℝ). Further consequences include quantitative approximation, preservation of spatial Bohr almost periodicity and its frequency module, conjugacy of spatial translation hulls, and preservation of the full complex Dirac spectrum and spatial transfer growth rates.
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BibTeX
@misc{kotani2026global,
title = {{Global flows for the cubic nonlinear Schrödinger equation without decay assumptions}},
author = {Shinichi Kotani and Yiqian Wang and Jiahao Xu and Shuo Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23324},
eprint = {2609.23324},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23324},
}
RIS
TY - UNPB
TI - Global flows for the cubic nonlinear Schrödinger equation without decay assumptions
AU - Shinichi Kotani
AU - Yiqian Wang
AU - Jiahao Xu
AU - Shuo Zhang
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23324
UR - https://arxiv.org/abs/2609.23324
AB - For each positive integer N, we construct global spectral group actions for the defocusing and focusing nonlinear Schrödinger hierarchies on initial-data classes containing W^2N+1,∞(ℝ). For N≥2, the quadratic flow gives global classical cubic NLS solutions. The construction extends the Sato--Segal--Wilson and Kotani frameworks to matrix Dirac systems; a homogeneous tau-function identity supplies the focusing positivity. No decay, smallness, periodicity, arithmetic condition, or prescribed spectral background is required for the initial-data inclusion. As a principal application, we prove global well-posedness on M_∞,1^5(ℝ) for both signs, with finite-time bounds and locally Lipschitz dependence on initial-data norm balls. The passage to this Banach-space flow uses near-real-axis Weyl asymptotics and continuity of normalized Toeplitz operators on compact initial-data sets to obtain uniform spatial bounds from five bounded derivatives. In the defocusing case, the flow extends to M_∞,1^5(ℝ)+H^1(ℝ). Further consequences include quantitative approximation, preservation of spatial Bohr almost periodicity and its frequency module, conjugacy of spatial translation hulls, and preservation of the full complex Dirac spectrum and spatial transfer growth rates.
ER -