关于Whitham型方程极端解的精确尖点行为
On the precise cusped behaviour of extreme solutions to Whitham-type equations
AI总结：
本文证明了两类非局部方程（包括单向和双向Whitham方程的最高稳态尖峰波）非平凡解在原点的精确首阶渐近行为，部分解决了相关猜想，且方法可推广至其他非局部方程并确定任意阶导数的渐近行为。
摘要：
我们证明了两个非局部方程族的非平凡解在原点的精确首阶渐近行为。所研究的方程包括单向和双向Whitham方程的最高稳态尖峰波所满足的方程。因此，该问题类似于捕捉经典最高Stokes波波峰处$120^\text{o}$内角的问题。特别地，我们的结果部分解决了最近一系列论文中针对此类极端波提出的猜想。我们的方法可以推广到其他非局部方程的解，并且还可用于确定其任意阶导数的渐近行为。
英文摘要：
We prove exact leading-order asymptotic behaviour at the origin for nontrivial solutions of two families of nonlocal equations. The equations investigated include those satisfied by the cusped highest steady waves for both the uni- and bidirectional Whitham equations. The problem is therefore analogous to that of capturing the $120\unicode{xb0}$ interior angle at the crests of classical Stokes' waves of greatest height. In particular, our results partially settle conjectures for such extreme waves posed in a series of recent papers. Our methods may be generalised to solutions of other nonlocal equations, and can moreover be used to determine asymptotic behaviour of their derivatives to any order.
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