Gravity and Capillary-Gravity Periodic Travelling Waves for Two Superposed Fluid Layers, One Being of Infinite Depth
Gravity and Capillary-Gravity Periodic Travelling Waves for Two Superposed Fluid Layers, One Being of Infinite Depth
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DOI：
10.1007/s000210050003
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发表时间：
1999-04-01
影响因子：
1.3
通讯作者：
Iooss, Gerard
中科院分区：
文献类型：
--
作者：
Iooss, Gerard
在两个完全流体的叠加层的势流中的二维行波的数学研究，具有自由表面和界面（有或没有表面张力），并且具有无限深度的底层，被设置为不适定的可逆演化问题，其中水平空间变量起着“时间”的作用。给出了线性化算子在平衡点附近的谱结构。这个谱包含一组孤立的本征值的有限重数，其中一小部分位于虚轴附近或虚轴上，整个真实的轴构成本质谱，其中没有本征值，除了在某些情况下为0。我们给出了一个一般的建设性证明分叉周期波，适应Lyapunov-施密特方法本（可逆）的情况下，0（这是“共振”）属于连续谱。特别是，我们给出了一般的情况下，1：1共振的情况下的结果。
The mathematical study of 2D travelling waves in the potential flow of two superposed layers of perfect fluid, with free surface and interfaces (with or without surface tensions) and with the bottom layer of infinite depth, is set as an ill-posed reversible evolution problem, where the horizontal space variable plays the role of a "time". We give the structure of the spectrum of the linearized operator near equilibrium. This spectrum contains a set of isolated eigenvalues of finite multiplicities, a small number of which lie near or on the imaginary axis, and the entire real axis constitutes the essential spectrum, where there is no eigenvalue, except 0 in some cases. We give a general constructive proof of bifurcating periodic waves, adapting the Lyapunov-Schmidt method to the present (reversible) case where 0 (which is "resonant") belongs to the continuous spectrum. In particular we give the results for the generic case and for the 1 : 1 resonance case.