The Regularity and Local Bifurcation of¶Steady Periodic Water Waves
The Regularity and Local Bifurcation of¶Steady Periodic Water Waves
复制标题
DOI：
10.1007/s002050000086
复制
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：
--
作者：
B. Buffoni;E. N. Dancer;J. Toland
用一个相当整洁的2π实变量周期函数的非线性伪微分算子方程的解，给出了在有或没有表面张力的自由表面上精确满足运动边界条件和动力边界条件的无限深度上的稳定周期水波。作为欧拉-拉格朗日方程，该公式具有梯度结构的优点，但由于它涉及非局部算子即希尔伯特变换，并且是拟线性的，因此比较复杂。本文是对所讨论的方程的数学研究。首先证明了其解是实解析解。然后利用梯度算子的分岔理论证明了在线性化问题的每一个特征值附近（与多重性无关）存在（非零）小振幅波。最后证明，当表面张力不存在时，从平凡解分叉的Stokes波分支的起始处不存在次谐波分岔或拐点。
Steady periodic water waves on infinite depth, satisfying exactly the kinematic and dynamic boundary conditions on the free surface, with or without surface tension, are given by solutions of a rather tidy nonlinear pseudo-differential operator equation for a 2π-periodic function of a real variable. Being an Euler-Lagrange equation, this formulation has the advantage of gradient structure, but is complicated by the fact that it involves a non-local operator, namely the Hilbert transform, and is quasi-linear.This paper is a mathematical study of the equation in question. First it is shown that itsW1,2solutions are real analytic. Then bifurcation theory for gradient operators is used to prove the existence of (non-zero) small amplitude waves near every eigenvalue (irrespective of multiplicity) of the linearised problem. Finally it is shown that when surface tension is absent there are no sub-harmonic bifurcations or turning points at the outset of the branches of Stokes waves which bifurcate from the trivial solution.