发表机构
Ningbo University; University of Science and Technology of China（宁波大学; 中国科学技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究一维含Dzyaloshinskii-Moriya相互作用的薛定谔映射中畴壁的渐近稳定性，证明其模平移旋转对称性稳定，辐射以t^{-1/2}衰减并具修正散射。
AI 中文摘要
本文研究了在加权Sobolev空间中，具有常数Dzyaloshinskii-Moriya相互作用的一维薛定谔映射的静态畴壁的小扰动的长时间动力学。扰动方程包含导数非线性，其线性化算子具有阈值共振。结果有两方面：（i）我们证明了畴壁在平移和旋转对称性模下的渐近稳定性。更精确地说，相应的调制参数随时间趋于无穷而收敛，而辐射以锐利速率$t^{-1/2}$衰减。（ii）我们证明了辐射表现出修正散射，具有由长程三次相互作用产生的显式对数相位修正。该证明发展了几何约化以克服导数损失，并利用了非线性相互作用中的结构相消。
英文摘要
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.
Comments63 pages
