Kuramoto模型中向部分锁定态的朗道阻尼
Landau damping to partially locked states in the Kuramoto model
AI总结：
该研究针对Kuramoto模型中部分锁定态的稳定性难题，构建含奇异测度傅里叶变换的函数空间，证明了弱拓扑下的局部渐近稳定性与显式谱判据，首次给出无耗散异质非规则平衡态的朗道阻尼证明。
摘要：
在全局耦合振子的Kuramoto模型中，部分锁定态（PLS）是一类定常解，体现了相互作用强度增加时部分同步性的涌现。尽管PLS早已被研究，但现有关于其稳定性的结果仅限于强拓扑下线性化动力学的中性稳定性，或是在通过所谓Ott-Antonsen（OA）假设得到的特定不变子空间中、针对振子特定频率分布的稳定性结论。在平均场极限下，Kuramoto模型展现出Vlasov方程中朗道阻尼机制的多种特征，这一类比一直是常规Kuramoto平衡态稳定性证明的灵感来源。此外，PLS渐近稳定性的主要数学难点在于这些态由异质奇异测度构成。本文针对一大类解析频率边缘分布，建立了PLS谱稳定性的显式判据，并证明了其在弱拓扑下的局部渐近稳定性。证明过程关键依赖于一个合适的函数空间，该空间包含奇异测度的傅里叶变换，且线性化动力学在其中能得到良好控制。为说明结论，我们在若干标准示例中计算了稳定性判据，特别证明了假设OA ansatz并不失一般性。据我们所知，该结果首次证明了无耗散情形下向异质非规则平衡态的朗道阻尼。
英文摘要：
In the Kuramoto model of globally coupled oscillators, partially locked states (PLS) are stationary solutions that incorporate the emergence of partial synchrony when the interaction strength increases. While PLS have long been considered, existing results on their stability are limited to neutral stability of the linearized dynamics in strong topology, or to specific invariant subspaces (obtained via the so-called Ott-Antonsen (OA) ansatz) with specific frequency distributions for the oscillators. In the mean field limit, the Kuramoto model shows various ingredients of the Landau damping mechanism in the Vlasov equation. This analogy has been a source of inspiration for stability proofs of regular Kuramoto equilibria. Besides, the major mathematical issue with PLS asymptotic stability is that these states consist of heterogeneous and singular measures. Here, we establish an explicit criterion for their spectral stability and we prove their local asymptotic stability in weak topology, for a large class of analytic frequency marginals. The proof strongly relies on a suitable functional space that contains (Fourier transforms of) singular measures, and for which the linearized dynamics is well under control. For illustration, the stability criterion is evaluated in some standard examples. We show in particular that no loss of generality results in assuming the OA ansatz. To our best knowledge, our result provides the first proof of Landau damping to heterogeneous and irregular equilibria, in absence of dissipation.
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