论文标题
Whitham方程的孤立波的全球分叉
Global bifurcation of solitary waves for the Whitham equation
论文作者
论文摘要
Whitham方程是一种非本地浅水波模型，将KDV方程的二次非线性与完整水波问题的线性分散体结合在一起。 Whitham猜想了Ehrnström和Wahlén的定期案例中，在周期性的情况下验证了最高，垂涎的波动溶液的存在。在本文中，我们证明了这是孤立波。与周期案例一样，证明基于全球分叉理论，但面临一些新的挑战。特别是，小振幅极限是奇异的，无法使用常规分叉理论来处理。取而代之的是，我们使用基于中心歧管定理的非本地版本的方法。在大振幅理论中，新的挑战是可能的紧凑性丧失，我们使用方程的定性特性排除了。最高波是作为全局分叉曲线的极限点。
The Whitham equation is a nonlocal shallow water-wave model which combines the quadratic nonlinearity of the KdV equation with the linear dispersion of the full water wave problem. Whitham conjectured the existence of a highest, cusped, traveling-wave solution, and his conjecture was recently verified in the periodic case by Ehrnström and Wahlén. In the present paper we prove it for solitary waves. Like in the periodic case, the proof is based on global bifurcation theory but with several new challenges. In particular, the small-amplitude limit is singular and cannot be handled using regular bifurcation theory. Instead we use an approach based on a nonlocal version of the center manifold theorem. In the large-amplitude theory a new challenge is a possible loss of compactness, which we rule out using qualitative properties of the equation. The highest wave is found as a limit point of the global bifurcation curve.
