The Cauchy problem for Manton's Chern-Simons-Schrödinger equation
Abstract
The Chern-Simons-Schrödinger equation (in the temporal gauge) arises as the Hamiltonian flow of the abelian Higgs energy on ℝ2. Manton (arXiv:hep-th/9701027) introduced the equation as a model for the dynamics of the critical points of the energy, known as vortices. He conjectured that, for a certain range of coupling constants, the vortex motion under the Chern-Simons-Schrödinger flow can be effectively captured by a first-order ODE on the moduli space of self-dual vortices constructed by Jaffe-Taubes (1980) and Samols (1992). As a first step towards a rigorous proof of Manton's conjecture, we formulate the Cauchy problem in DeTurck gauge within the natural energy space and prove global well-posedness. We also obtain, as a corollary of the well-posedness theory and the stability results in our previous work (arXiv:2603.24900), orbital stability of the self-dual vortices under the Chern-Simons-Schrödinger flow near self-dual coupling. The heart of our analysis lies in developing a geometric Littlewood-Paley theory based on the covariant heat equation in caloric gauge, which we use to perform a paradifferential-style decomposition of the Chern-Simons-Schrödinger equation.
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BibTeX
@misc{zhao2026cauchy,
title = {{The Cauchy problem for Manton's Chern-Simons-Schrödinger equation}},
author = {Jason Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13096},
eprint = {2609.13096},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13096},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Cauchy problem for Manton's Chern-Simons-Schrödinger equation
AU - Jason Zhao
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13096
UR - https://arxiv.org/abs/2609.13096
AB - The Chern-Simons-Schrödinger equation (in the temporal gauge) arises as the Hamiltonian flow of the abelian Higgs energy on ℝ^2. Manton (arXiv:hep-th/9701027) introduced the equation as a model for the dynamics of the critical points of the energy, known as vortices. He conjectured that, for a certain range of coupling constants, the vortex motion under the Chern-Simons-Schrödinger flow can be effectively captured by a first-order ODE on the moduli space of self-dual vortices constructed by Jaffe-Taubes (1980) and Samols (1992). As a first step towards a rigorous proof of Manton's conjecture, we formulate the Cauchy problem in DeTurck gauge within the natural energy space and prove global well-posedness. We also obtain, as a corollary of the well-posedness theory and the stability results in our previous work (arXiv:2603.24900), orbital stability of the self-dual vortices under the Chern-Simons-Schrödinger flow near self-dual coupling. The heart of our analysis lies in developing a geometric Littlewood-Paley theory based on the covariant heat equation in caloric gauge, which we use to perform a paradifferential-style decomposition of the Chern-Simons-Schrödinger equation.
ER -