发表机构
China University of Mining and Technology（中国矿业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对 Sawada-Kotera 方程，通过新的亚纯 Riemann-Hilbert 框架和 \\(\bar{\partial}\\)-非线性最速下降法，证明了孤子分解猜想，建立了 N-孤子解的渐近稳定性，并扩展到孤子与辐射共存的情形。
AI 中文摘要
我们推导了具有连续谱和有限离散谱的可容许数据的 Sawada-Kotera (SK) 方程在不同时空区域的长时渐近行为。我们的结果证明了孤子分解猜想，给出了由 F-XVIII Painlevé 超越函数控制的渐近行为，并建立了 N-孤子解的渐近稳定性。更重要的是，这将长时渐近理论扩展到孤子与辐射共存的情形，而此前的结果仅限于纯连续谱。我们论证中的一个关键要素是新的亚纯 Riemann-Hilbert 框架，该框架通过完整的六点极点轨道纳入离散谱，同时保持余因子解析性和谱原点处的奇异结构。第二个要素是将逐轨道、区域相关的极点约简机制与 \\(\bar{\partial}\\)-非线性最速下降法相结合，这使我们能够选择每个区域中贡献的孤子并将其与连续辐射分离。我们的框架还通过正则修正 SK 问题和 Miura 映射解决了奇异谱原点处会合的驻点问题。
英文摘要
We derive long-time asymptotics in different space-time regions for the Sawada-Kotera (SK) equation with admissible data involving continuous spectrum and finite discrete spectrum. Our result proves the soliton resolution conjecture, yields the asymptotics governed by the F-XVIII Painlevé transcendent, and establishes the asymptotic stability of \(N\)-soliton solutions. More importantly, this extends the long-time asymptotic theory to the coexistence of solitons and radiation, whereas previous results were restricted to the purely continuous spectrum. One key ingredient in our argument is a new meromorphic Riemann-Hilbert framework that incorporates the discrete spectrum through complete six-point pole orbits while preserving cofactor analyticity and the singular structure at the spectral origin. A second ingredient is the combination of an orbitwise, region-dependent pole reduction mechanism and the \(\bar{\partial}\)-nonlinear steepest descent method, which allows us to select the solitons contributing in each region and separate them from the continuous radiation. Our framework also resolves the coalescing stationary points at the singular spectral origin through the regular modified SK problem and the Miura map.
Comments61 pages, 12 figures
