关于能量空间中 sine-Gordon kink 的渐近稳定性
On the asymptotic stability of the sine-Gordon kink in the energy space
AI总结：
本文研究了 sine-Gordon 方程中共振对 kink 渐近稳定性的影响，利用 Bäcklund 变换和 Virial 估计构造了接近 kink 的初始数据光滑流形，证明了其能量空间渐近稳定性，并揭示了 breathers 与 wobbling kinks 的联系。
摘要：
我们考虑 1+1 维中的 sine-Gordon (SG) 方程。kink 是 SG 的静态、非对称精确解，在能量空间 $H^1\ imes L^2$ 中是稳定的。众所周知，kink 附近的线性化算子具有简单的核且无内部模式。然而，它在连续谱底部拥有一个奇共振，这与著名的 wobbling kink 的存在密切相关。wobbling kink 是 SG 围绕 kink 的显式时间周期解，它违背了能量空间中 kink 的渐近稳定性。在本文中，我们进一步研究了共振在渐近稳定性问题中的影响。我们还讨论了 SG 背景下 breathers、wobbling kinks 和共振之间的关系。通过汇集 Bäcklund 变换（BT）和围绕真空解奇扰动的 Virial 估计，我们首先确定了零点附近的初始数据流形，在该流形下 BT 与 wobbling kink 解相关。结果表明，（偶）小 breathers 与 kink 附近的奇扰动（包括 wobbling kink 本身）密切相关。作为该结果的推论，利用 BT 我们可以构造一个接近 kink 的初始数据光滑流形，对其而言在能量空间中存在渐近稳定性。该初始数据具有 (kink + 奇, 偶) 形式的空间对称性，原则上非共振，且不被流保持。尽管 SG 中存在 wobbling kinks，该渐近稳定性依然成立。我们还证明了 wobbling kinks 在奇数据下是轨道稳定的，并阐明了 SG 和 $ϕ^4$ 在线性 Bäcklund 变换层面的一些有趣联系。
英文摘要：
We consider the sine-Gordon (SG) equation in 1+1 dimensions. The kink is a static, non-symmetric exact solution to SG, stable in the energy space $H^1\times L^2$. It is well-known that the linearized operator around the kink has a simple kernel and no internal modes. However, it possesses an odd resonance at the bottom of the continuum spectrum, deeply related to the existence of the (in)famous wobbling kink, an explicit periodic-in-time solution of SG around the kink that contradicts the asymptotic stability of the kink in the energy space. In this paper, we further investigate the influence of resonances in the asymptotic stability question. We also discuss the relationship between breathers, wobbling kinks and resonances in the SG setting. By gathering Bäcklund transformations (BT) and Virial estimates around odd perturbations of the vacuum solution, we first identify the manifold of initial data around zero under which BTs are related to the wobbling kink solution. It turns out that (even) small breathers are deeply related to odd perturbations around the kink, including the wobbling kink itself. As a consequence of this result, using BTs we can construct a smooth manifold of initial data close to the kink, for which there is asymptotic stability in the energy space. The initial data has spatial symmetry of the form (kink + odd, even), non-resonant in principle, and not preserved by the flow. This asymptotic stability property holds despite the existence of wobbling kinks in SG. We also show that wobbling kinks are orbitally stable under odd data, and clarify some interesting connections between SG and $ϕ^4$ at the level of linear Bäcklund transformations.
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