Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves
Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves
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无限深度的无粘性流体的自由表面上的毛细管重力波。
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
Pius Kirrmann
中科院分区：
文献类型：
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作者：
G. Iooss;Pius Kirrmann
研究了无限深二维无粘流体自由表面上的永久毛细重力波。速端图变换的应用将自由边值问题转化为具有非线性微分边界条件的下半平面中的Cauchy-Riemann方程的边值问题。这可以转换为符号为−k2+4的积分微分方程|K| −4（1+μ），其中μ是分叉参数。本文给出了一个规范形式的分析，它表明边值问题可以化为一个可积的常微分方程组加上一个含有高阶非局部项的余项，|μ|小了这个规范形系统已经被几位作者（Looss &Kirchgässner [8]，Looss &Pérouème [10]，Dias & Looss [5]）深入研究过。它存在一对在Kirchgässner [11]意义下可逆的孤波解。应用文[11]中介绍的方法，证明了这对可逆孤立波对边值问题是持续存在的，并且这对孤立波在无穷远处的衰减至少为1/|X|.
Permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated. An application of the hodograph transform converts the free boundary-value problem into a boundary-value problem for the Cauchy-Riemann equations in the lower halfplane with nonlinear differential boundary conditions. This can be converted to an integro-differential equation with symbol −k2+4|k|−4(1+μ), where μ is a bifurcation parameter. A normal-form analysis is presented which shows that the boundary-value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for |μ| small. This normal form system has been studied thoroughly by several authors (Iooss &Kirchgässner [8],Iooss &Pérouème [10],Dias &Iooss [5]). It admits a pair of solitary-wave solutions which are reversible in the sense ofKirchgässner [11]. By applying a method introduced in [11], it is shown that this pair of reversible solitary waves persists for the boundary-value problem, and that the decay at infinity of these solitary waves is at least like 1/|x|.