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Asymptotic behavior analysis of solutions to the Heisenberg ferromagnet equation with the Schwartz initial data

  • Xumeng Zhou
  • Xianguo Geng
  • Bo Xue
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Abstract

This work aims to investigate the long-time asymptotic behavior of solutions to the Cauchy problem for the Heisenberg ferromagnet equation with the Schwartz initial data. Utilizing the gauge transformations, spectral analysis and the inverse scattering method, we prove that the solutions to the Heisenberg ferromagnet equation can be expressed in terms of the solutions to a matrix Riemann-Hilbert problem formulated in the complex k-plane. Various Deift-Zhou contour deformations and the motivation behind them are given. By applying the nonlinear steepest descent method to the associated matrix-valued Riemann-Hilbert problem, we obtain the exact leading-order asymptotic formulas and uniform error estimates for solutions to the Cauchy problem of the Heisenberg ferromagnet equation.

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Cite

BibTeXRIS
BibTeX
@misc{zhou2026asymptotic,
  title = {{Asymptotic behavior analysis of solutions to the Heisenberg ferromagnet equation with the Schwartz initial data}},
  author = {Xumeng Zhou and Xianguo Geng and Bo Xue},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37573},
  eprint = {2609.37573},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37573},
}
RIS
TY  - UNPB
TI  - Asymptotic behavior analysis of solutions to the Heisenberg ferromagnet equation with the Schwartz initial data
AU  - Xumeng Zhou
AU  - Xianguo Geng
AU  - Bo Xue
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37573
UR  - https://arxiv.org/abs/2609.37573
AB  - This work aims to investigate the long-time asymptotic behavior of solutions to the Cauchy problem for the Heisenberg ferromagnet equation with the Schwartz initial data. Utilizing the gauge transformations, spectral analysis and the inverse scattering method, we prove that the solutions to the Heisenberg ferromagnet equation can be expressed in terms of the solutions to a matrix Riemann-Hilbert problem formulated in the complex k-plane. Various Deift-Zhou contour deformations and the motivation behind them are given. By applying the nonlinear steepest descent method to the associated matrix-valued Riemann-Hilbert problem, we obtain the exact leading-order asymptotic formulas and uniform error estimates for solutions to the Cauchy problem of the Heisenberg ferromagnet equation.
ER  -