The Sub-Harmonic Bifurcation of Stokes Waves
The Sub-Harmonic Bifurcation of Stokes Waves
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DOI：
10.1007/s002050000087
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发表时间：
2000-06
影响因子：
2.5
通讯作者：
B. Buffoni;E. N. Dancer;J. Toland
中科院分区：
文献类型：
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作者：
B. Buffoni;E. N. Dancer;J. Toland
在重力作用下无表面张力的无限深无旋流自由表面上的稳定周期水波（Stokes波）可以用包含希尔伯特变换的拟线性方程的解来描述，该方程是简单泛函的欧拉-拉格朗日方程。未知数是一个2π周期函数，它给出了波的轮廓和弗劳德数，弗劳德数是一个反映波速固定时波长的无量纲参数（反之亦然）。虽然这个方程是精确的，但它是二次的（没有高阶项），它的解集的整体结构可以用实解析变分理论和变分技术的元素来研究。本文证明了从线性化问题的第一个特征值衍生出一个具有规定最小周期的唯一定义的圆弧连通解集，尽管它不一定是最大的连通解集，也可能是自相交的，但它具有一个局部实解析参数化，并且在其闭包（适当定义）中包含一个最大高度的波。此外，它包含无穷多个点，这些点要么是转折点，要么是具有规定最小周期的解分叉的点。（数字证据表明，只有前者发生，这仍然是一个悬而未决的问题。）还证明了存在无穷多个弗劳德数值，Stokes波的最小波长是基本波长的任意大整数倍，从初级分支分叉。这些是论文标题中的次谐波分岔。(1925年，列维-奇维塔推测，以速度传播的斯托克斯波的最小波长不超过2πc2/g。我们在次谐波分岔上的结果反驳了这一点，因为它表明存在传播速度有限但最小波长任意大的斯托克斯波。尽管Benjamin & Feir}和其他人[9,10]的工作表明深水中的斯托克斯波是不稳定的，但它们在理论流体动力学中仍然占有中心地位。本文研究它们的数学工具是实解析函数理论、周期线性伪微分算子的谱理论和莫尔斯理论，并结合了Plotnikov[36]的一篇论文的深刻影响。
Steady periodic water waves on the free surface of an infinitely deep irrotational flow under gravity without surface tension (Stokes waves) can be described in terms of solutions of a quasi-linear equation which involves the Hilbert transform and which is the Euler-Lagrange equation of a simple functional. The unknowns are a 2π-periodic functionwwhich gives the wave profile and the Froude number, a dimensionless parameter reflecting the wavelength when the wave speed is fixed (andvice versa).Although this equation is exact, it is quadratic (with no higher order terms) and the global structure of its solution set can be studied using elements of the theory of real analytic varieties and variational techniques.In this paper it is shown that there bifurcates from the first eigenvalue of the linearised problem a uniquely defined arc-wise connected set of solutions with prescribed minimal period which, although it is not necessarily maximal as a connected set of solutions and may possibly self-intersect, has a local real analytic parametrisation and contains a wave of greatest height in its closure (suitably defined). Moreover it contains infinitely many points which are either turning points or points where solutions with the prescribed minimal period bifurcate. (The numerical evidence is that only the former occurs, and this remains an open question.)It is also shown that there are infinitely many values of the Froude number at which Stokes waves, having a minimal wavelength that is an arbitrarily large integer multiple of the basic wavelength, bifurcate from the primary branch. These are the sub-harmonic bifurcations in the paper's title. (In 1925 Levi-Civita speculated that the minimal wavelength of a Stokes wave propagating with speedcdid not exceed 2πc2/g. This is disproved by our result on sub-harmonic bifurcation, since it shows that there are Stokes waves with bounded propagation speeds but arbitrarily large minimal wavelengths.)Although the work of Benjamin & Feir} and others [9, 10] has shown Stokes waves on deep water to be unstable, they retain a central place in theoretical hydrodynamics. The mathematical tools used to study them here are real analytic-function theory, spectral theory of periodic linear pseudo-differential operators and Morse theory, all combined with the deep influence of a paper by Plotnikov [36].