发表机构
Ulsan National Institute of Science and Technology（蔚山科学技术院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文以Camassa-Holm方程为例，证明光滑初值经波浪破碎后首次奇点立即分叉为反尖峰-尖峰对，并给出其正则性与动力学机制。
AI 中文摘要
我们研究了通过波浪破碎出现尖峰对的现象，以Camassa-Holm方程为主要模型。对于一大类开集内的光滑初值，首次奇点出现在唯一一点，并具有尖锐的局部Hölder正则性C^{3/5}。我们构造了通过该奇异时刻的拉格朗日流的延拓，并从中重构出一个保守的欧拉弱解。我们表明，首次奇点并不会在破碎后简单地持续存在；相反，它立即分叉为一对反尖峰-尖峰，每个都具有尖锐的局部Hölder正则性C^{2/3}，且具有相反的单侧斜率方向。这两个奇异分支从原始破碎点分离移动，它们之间的剖面部分反转了其局部单调性。证明基于一个扩展的逆斜率变量，该变量在波浪破碎过程中保持正则。在首次奇异时刻，该变量具有非退化的二次零点，随后立即分裂为两个简单零点。拉格朗日流映射的相应退化将二次的首次破碎几何转换为C^{3/5}剖面，并将破碎后的简单零点几何转换为C^{2/3}尖峰剖面。广义Hunter-Saxton族提供了同一拉格朗日机制的显式实现，并允许我们跟踪奇异分支随后的相互作用和湮灭。因此，反尖峰-尖峰对通过波浪破碎从光滑初值动态出现，而不是通过奇异行波假设强加的。
英文摘要
We study the emergence of cuspons through wave breaking, with the Camassa--Holm equation as a principal model. For a broad open class of smooth initial data, the first singularity occurs at a unique point and has the sharp local Hölder regularity \(C^{3/5}\). We construct a continuation of the Lagrangian flow through this singular time and reconstruct from it a conservative Eulerian weak solution. We show that the first singularity does not simply persist after breaking; instead, it bifurcates immediately into an anticuspon--cuspon pair, each having the sharp local Hölder regularity \(C^{2/3}\), with opposite one-sided slope orientations. The two singular branches move apart from the original breaking point, and the portion of the profile between them reverses its local monotonicity. The proof is based on an extended inverse-slope variable that remains regular through wave breaking. At the first singular time, this variable has a nondegenerate quadratic zero, which splits immediately afterward into two simple zeros. The corresponding degeneracy of the Lagrangian flow map converts the quadratic first-breaking geometry into the \(C^{3/5}\) profile and the post-breaking simple-zero geometry into the \(C^{2/3}\) cuspon profiles. The generalized Hunter--Saxton family provides an explicit realization of the same Lagrangian mechanism and also allows us to track the subsequent interaction and annihilation of the singular branches. Thus an anticuspon--cuspon pair emerges dynamically from smooth initial data through wave breaking, rather than being imposed through a singular traveling-wave ansatz.
Comments42 pages, 4 figures
