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Sharp non-uniqueness of singular solutions to the one-dimensional periodic cubic nonlinear Schrödinger equation
Abstract
We construct by convex integration a nonzero singular weak solution of the one-dimensional periodic cubic nonlinear Schrödinger equation that is compactly supported in time, has zero initial datum, and satisfies u∈⋂α<1/6 Ct0 Hxα∩ ⋂1≤ p<3 Ct0 Lxp. The key new ingredient is a two-carrier perturbation: its zero-output interaction cancels the previous error, without introducing any zero Fourier mode, which is crucial in the cubic NLS. Whenever a singular solution belongs to Ct0Lx3, this nonlinearity agrees with the ordinary product |u|2u. Together with unconditional uniqueness at H1/6, this makes the threshold sharp within the singular solution class considered here.
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BibTeX
@misc{peng2026sharp,
title = {{Sharp non-uniqueness of singular solutions to the one-dimensional periodic cubic nonlinear Schrödinger equation}},
author = {Qirui Peng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.07866},
eprint = {2608.07866},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.07866},
}
RIS
TY - UNPB
TI - Sharp non-uniqueness of singular solutions to the one-dimensional periodic cubic nonlinear Schrödinger equation
AU - Qirui Peng
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2608.07866
UR - https://arxiv.org/abs/2608.07866
AB - We construct by convex integration a nonzero singular weak solution of the one-dimensional periodic cubic nonlinear Schrödinger equation that is compactly supported in time, has zero initial datum, and satisfies u∈⋂_α<1/6 C_t^0 H_x^α∩ ⋂_1≤ p<3 C_t^0 L_x^p. The key new ingredient is a two-carrier perturbation: its zero-output interaction cancels the previous error, without introducing any zero Fourier mode, which is crucial in the cubic NLS. Whenever a singular solution belongs to C_t^0L_x^3, this nonlinearity agrees with the ordinary product |u|^2u. Together with unconditional uniqueness at H^1/6, this makes the threshold sharp within the singular solution class considered here.
ER -