论文标题
不稳定的stokes波
Unstable Stokes waves
论文作者
论文摘要
我们研究了$2π/κ$周期性振幅波的光谱不稳定，在重力下以单位深度的水为单位。数值证据表明，每当不受干扰的波与其无限扰动共鸣时，不稳定。除了在复合平面的起源附近的benjamin- feir不稳定性外，尚未对此进行分析研究。在这里，我们开发了一种定期的Evans功能方法，以提供本杰明（Benjamin）的替代证明 - 福克斯（Feir）的不稳定性，还提供了远离原点的光谱不稳定的第一个证明。具体而言，我们证明了$κ>κ_1的原点附近的不稳定性：= 1.3627827 \点$ $ and dots $ and dots $ and Inderability wy dots $ toter的共鸣，只要索引函数为正，我们就会出现订单。经过验证的数字确定，指数函数确实对某些$κ<κ_1$呈阳性，在这种情况下，即使Benjamin- feir不稳定不舒服，它也存在频谱上不稳定的stokes波。证据涉及中心流形的降低，浮点理论以及普通和部分微分方程的方法。数值评估表明，除非$κ= 1.8494040 \ dots $，否则指数函数仍为正。因此，我们猜想所有振幅足够小的stoke波在光谱上都是不稳定的。为了证明猜想，必须验证指数函数对$κ$足够小的阳性。
We investigate the spectral instability of a $2π/κ$ periodic Stokes wave of sufficiently small amplitude, traveling in water of unit depth, under gravity. Numerical evidence suggests instability whenever the unperturbed wave is resonant with its infinitesimal perturbations. This has not been analytically studied except for the Benjamin--Feir instability in the vicinity of the origin of the complex plane. Here we develop a periodic Evans function approach to give an alternative proof of the Benjamin--Feir instability and, also, a first proof of spectral instability away from the origin. Specifically, we prove instability near the origin for $κ>κ_1:=1.3627827\dots$ and instability due to resonance of order two so long as an index function is positive. Validated numerics establishes that the index function is indeed positive for some $κ<κ_1$, whereby there exists a Stokes wave that is spectrally unstable even though it is insusceptible to the Benjamin--Feir instability. The proofs involve center manifold reduction, Floquet theory, and methods of ordinary and partial differential equations. Numerical evaluation reveals that the index function remains positive unless $κ=1.8494040\dots$. Therefore, we conjecture that all Stokes waves of sufficiently small amplitude are spectrally unstable. For the proof of the conjecture, one has to verify that the index function is positive for $κ$ sufficiently small.
