Sharp well-posedness and soliton resolution for the half-wave maps equation in H1/2•
Abstract
We prove sharp well-posedness and soliton resolution for the half-wave maps equation on the real line R in H1/2•. This regularity threshold for global well-posedness is optimal, as we show by constructing examples of ill-posedness in Hs• for any s<1/2. Moreover, with the spectral assumption satisfied by a dense set of initial data in the energy space, we prove that solutions in H1/2• asymptotically decouple as an infinite superposition of solitons. Our results extend the known global well-posedness and finite soliton resolution for rational data in H∞• to sharp global well-posedness and soliton resolution for general data in H1/2•. Different from rational solutions, general data can also involve an infinite discrete spectrum, which makes the reduction analysis framework on the finite rank no longer applicable. For sharp well-posedness, we present a new isometry criterion and stability results for resolvent transform in the explicit formula. For soliton resolution, we develop a spectral method for this equation based on direct resolvent analysis and a novel trace norm estimate that upgrades energy convergence to uniform convergence. Our argument also yields sharp well-posedness and soliton resolution for the generalized half-wave maps equation with targets in the complex Grassmannians, extending the classical sphere-valued case.
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BibTeX
@misc{tang2026sharp,
title = {{Sharp well-posedness and soliton resolution for the half-wave maps equation in $H^1/2_\bullet$}},
author = {Fei Tang and Shou-Fu Tian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36676},
eprint = {2610.36676},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36676},
}
RIS
TY - UNPB
TI - Sharp well-posedness and soliton resolution for the half-wave maps equation in H^1/2_•
AU - Fei Tang
AU - Shou-Fu Tian
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36676
UR - https://arxiv.org/abs/2610.36676
AB - We prove sharp well-posedness and soliton resolution for the half-wave maps equation on the real line R in H^1/2_•. This regularity threshold for global well-posedness is optimal, as we show by constructing examples of ill-posedness in H^s_• for any s<1/2. Moreover, with the spectral assumption satisfied by a dense set of initial data in the energy space, we prove that solutions in H^1/2_• asymptotically decouple as an infinite superposition of solitons. Our results extend the known global well-posedness and finite soliton resolution for rational data in H^∞_• to sharp global well-posedness and soliton resolution for general data in H^1/2_•. Different from rational solutions, general data can also involve an infinite discrete spectrum, which makes the reduction analysis framework on the finite rank no longer applicable. For sharp well-posedness, we present a new isometry criterion and stability results for resolvent transform in the explicit formula. For soliton resolution, we develop a spectral method for this equation based on direct resolvent analysis and a novel trace norm estimate that upgrades energy convergence to uniform convergence. Our argument also yields sharp well-posedness and soliton resolution for the generalized half-wave maps equation with targets in the complex Grassmannians, extending the classical sphere-valued case.
ER -