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首页> 外文期刊>Journal of dynamics and differential equations >A Spatial Dynamics Theory for Doubly Periodic Travelling Gravity-Capillary Surface Waves on Water of Infinite Depth
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A Spatial Dynamics Theory for Doubly Periodic Travelling Gravity-Capillary Surface Waves on Water of Infinite Depth

机译：无限深度水上双周期行进的重力-毛细表面波的空间动力学理论

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摘要

In this article we show how Kirchgassner's spatial dynamics approach can be used to construct doubly periodic travelling gravity-capillary surface waves on water of infinite depth. The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable and the infinite-dimensional function space consists of wave profiles which are periodic (with fixed period) in a second, different horizontal direction. The imaginary part of the spectrum of the linearised Hamiltonian vector field consists of essential spectrum at the origin and a finite number of eigenvalues whose distribution is described geometrically. Periodic solutions to the spatial Hamiltonian system are detected using Iooss's generalisation of the reversible Lyapunov centre theorem; these solutions correspond to doubly periodic solutions to the travelling water-wave problem. For a generic choice of the periodic domain there exist values of the physical parameters at which a doubly periodic wave with this domain exists.
机译：在本文中，我们演示了如何使用Kirchgassner的空间动力学方法在无限深度的水上构造双周期行进的重力毛细管表面波。流体力学问题被表述为可逆的汉密尔顿系统，其中任意水平空间方向都是时变变量，而无穷维函数空间由在第二个不同水平方向上呈周期性（具有固定周期）的波轮廓组成。线性化哈密顿向量场的频谱的虚部由原点处的基本频谱和有限数量的特征值组成，这些特征值的分布用几何描述。使用Iooss对可逆Lyapunov中心定理的推广来检测空间哈密顿系统的周期解；这些解对应于行进水波问题的双周期解。对于周期域的一般选择，存在物理参数的值，在该物理参数处存在具有该域的双周期波。

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