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Asymptotic Stability of Domain Walls for One Dimensional Schrödinger Map with Dzyaloshinskii-Moriya Interaction

  • Ze Li
  • Zifan Wang
  • Lifeng Zhao
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Abstract

In this paper, we study the long time dynamics of small perturbations of static domain walls in weighted Sobolev spaces for the one dimensional Schrödinger map with constant Dzyaloshinskii-Moriya interaction. The perturbation equation contains derivative nonlinearities, and its linearized operator has a threshold resonance. The results are twofold. (i) We prove the asymptotic stability of the domain wall modulo the translation and rotation symmetries. More precisely, the corresponding modulation parameters converge as time tends to infinity, while the radiation decays at the sharp rate t−1/2. (ii) We prove that the radiation exhibits modified scattering, with an explicit logarithmic phase correction generated by the long range cubic interaction. The proof develops geometric reduction to overcome the loss of derivatives and exploits structural cancellations in the nonlinear interactions.

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Cite

BibTeXRIS
BibTeX
@misc{li2026asymptotic,
  title = {{Asymptotic Stability of Domain Walls for One Dimensional Schrödinger Map with Dzyaloshinskii-Moriya Interaction}},
  author = {Ze Li and Zifan Wang and Lifeng Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10458},
  eprint = {2610.10458},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10458},
}
RIS
TY  - UNPB
TI  - Asymptotic Stability of Domain Walls for One Dimensional Schrödinger Map with Dzyaloshinskii-Moriya Interaction
AU  - Ze Li
AU  - Zifan Wang
AU  - Lifeng Zhao
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10458
UR  - https://arxiv.org/abs/2610.10458
AB  - In this paper, we study the long time dynamics of small perturbations of static domain walls in weighted Sobolev spaces for the one dimensional Schrödinger map with constant Dzyaloshinskii-Moriya interaction. The perturbation equation contains derivative nonlinearities, and its linearized operator has a threshold resonance. The results are twofold. (i) We prove the asymptotic stability of the domain wall modulo the translation and rotation symmetries. More precisely, the corresponding modulation parameters converge as time tends to infinity, while the radiation decays at the sharp rate t^−1/2. (ii) We prove that the radiation exhibits modified scattering, with an explicit logarithmic phase correction generated by the long range cubic interaction. The proof develops geometric reduction to overcome the loss of derivatives and exploits structural cancellations in the nonlinear interactions.
ER  -