Time quasi-periodic gravity water waves in finite depth
Time quasi-periodic gravity water waves in finite depth
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DOI：
10.1007/s00222-018-0812-2
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发表时间：
2017-08
影响因子：
3.1
通讯作者：
P. Baldi;M. Berti;E. Haus;Riccardo Montalto
中科院分区：
文献类型：
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作者：
P. Baldi;M. Berti;E. Haus;Riccardo Montalto
在纯重力作用下，证明了有限深度二维海洋的小振幅时准周期驻水波解（即周期和空间偶变量）的Cantor族的存在性和线性稳定性。这个结果对渐近满测度的Borel集的所有深度参数值都成立。这是一个小除数问题。主要的困难是重力水波方程的完全非线性性质——最高阶导数出现在非线性项中，而不是在原点的线性化中——以及线性频率在无穷远处以次线性方式增长的事实。为了克服这些问题，我们首先利用准周期变量的伪微分变化，将沿Nash-Moser迭代格式在每个近似准周期解处得到的线性化算子简化为常系数直至平滑算子。然后我们应用了一个KAM约化方案，该方案需要在时间和空间上都失去导数的非常弱的Melnikov非共振条件。尽管深度参数对线性频率的移动只是指数级的小量，但我们能够验证大多数深度值的非共振条件，扩展简并的KAM理论。
We prove the existence and the linear stability of Cantor families of small amplitude timequasi-periodicstanding water wave solutions—namely periodic and even in the space variablex—of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the fully nonlinear nature of the gravity water waves equations—the highest orderx-derivative appears in the nonlinear term but not in the linearization at the origin—and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators, obtained at each approximate quasi-periodic solution along a Nash–Moser iterative scheme, to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions which lose derivatives both in time and space. Despite the fact that the depth parameter moves the linear frequencies by just exponentially small quantities, we are able to verify such non-resonance conditions for most values of the depth, extending degenerate KAM theory.