关于理想双纽线、蝴蝶与圆波
On ideal lemniscates, butterflies, and circular waves
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中文总结 AI 辅助
该研究证明长度惩罚理想能量存在无穷多类临界点，对双纽线、蝴蝶族用分阶段直接极小化法，圆波族用适配打靶法，还通过边界层分析明确两类菲涅耳相位对应的临界点，为相关几何问题提供了存在性依据。
中文摘要 AI 辅助
我们证明了长度惩罚理想能量存在无穷多个类双纽线、类蝴蝶与圆波的临界点。对于双纽线和蝴蝶族，我们采用分阶段直接极小化过程确立其存在性；边界层分析显示，在两个菲涅耳相位附近存在临界点：模2π为3π/4对应双纽线，模2π为7π/4对应蝴蝶。圆波族在某种意义上由多重覆盖的圆分岔产生，我们采用适配的打靶法确立其存在性。
英文摘要
We prove the existence of infinitely many lemniscate-like, butterfly-like, and circular-wave critical points for the length-penalised ideal energy. For the lemniscate and butterfly family, we use a staged direct minimisation process to establish existence. A boundary-layer analysis reveals critical points near two Fresnel phases: $3π/4$ modulo $2π$, corresponding to lemniscates, and $7π/4$ modulo $2π$, which are the butterflies. The family of circular waves bifurcates, in a sense, from multiply-covered circles. We use an adapted shooting method to establish their existence.
