论文标题
在重力下无限深流体上的准周期性流动波
Quasi-periodic traveling waves on an infinitely deep fluid under gravity
论文作者
论文摘要
我们考虑具有无限深度的周期性一维界面的重力水波系统，我们确定了小振幅的存在和线性稳定性，时代的准周期性，流动波。这提供了从\ emph {完全谐振}椭圆固定点分叉的准周期水波解决方案的第一个存在结果。该证明基于NASH-MOSER方案，Birkhoff正常形式方法和伪差分计算技术。我们处理\ emph {小除数}的组合问题和方程式的\ emph {Full-nonlinear}。缺乏参数，例如毛细血管或海洋深度，要求进行精制的\ emph {nonlialear}分叉分析，涉及几种非平凡的共振波相互作用，作为众所周知的“本杰明 - - 弗吉尔共鸣”。我们开发了一种新型的正常形式方法来处理这种方法。此外，通过充分利用哈密顿结构，我们能够提供一类宽类的解决方案，这些解决方案不受时间和空间变量的限制。
We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the first existence result of quasi-periodic water waves solutions bifurcating from a \emph{completely resonant} elliptic fixed point. The proof is based on a Nash-Moser scheme, Birkhoff normal form methods and pseudo-differential calculus techniques. We deal with the combined problems of \emph{small divisors} and the \emph{fully-nonlinear} nature of the equations. The lack of parameters, like the capillarity or the depth of the ocean, demands a refined \emph{nonlinear} bifurcation analysis involving several non-trivial resonant wave interactions, as the well-known "Benjamin-Feir resonances". We develop a novel normal form approach to deal with that. Moreover, by making full use of the Hamiltonian structure, we are able to provide the existence of a wide class of solutions which are free from restrictions of parity in the time and space variables.
