Global well-posedness of defocusing cubic NLS in M∞,1(ℝ)
Abstract
We prove global well-posedness of the one-dimensional defocusing cubic nonlinear Schrödinger equation in the modulation space M∞,1(ℝ). This space imposes no spatial decay and contains Cb2(ℝ) as well as all absolutely convergent sums of plane waves. The result applies to arbitrary data in this space, including large smooth quasiperiodic profiles and their localized perturbations. The proof constructs a nonnegative density satisfying a local conservation law from forward Weyl ratios, which are defined through half-line square-integrable solutions of the associated spectral problem. A suitable nonlinear combination of localized integrals of this density controls the modulation norm. Finally, choosing the spatial localization scale and NLS scaling in a coordinated way makes the accumulated boundary flux small enough to continue every mild solution globally.
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BibTeX
@misc{klaus2026global,
title = {{Global well-posedness of defocusing cubic NLS in $M^\infty,1(\mathbbR)$}},
author = {Friedrich Klaus},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.29147},
eprint = {2609.29147},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29147},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global well-posedness of defocusing cubic NLS in M^∞,1(ℝ)
AU - Friedrich Klaus
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2609.29147
UR - https://arxiv.org/abs/2609.29147
AB - We prove global well-posedness of the one-dimensional defocusing cubic nonlinear Schrödinger equation in the modulation space M^∞,1(ℝ). This space imposes no spatial decay and contains C_b^2(ℝ) as well as all absolutely convergent sums of plane waves. The result applies to arbitrary data in this space, including large smooth quasiperiodic profiles and their localized perturbations. The proof constructs a nonnegative density satisfying a local conservation law from forward Weyl ratios, which are defined through half-line square-integrable solutions of the associated spectral problem. A suitable nonlinear combination of localized integrals of this density controls the modulation norm. Finally, choosing the spatial localization scale and NLS scaling in a coordinated way makes the accumulated boundary flux small enough to continue every mild solution globally.
ER -