The existence of solitary water waves
The existence of solitary water waves
复制标题
DOI：
10.1002/cpa.3160300402
复制
发表时间：
1977-07
影响因子：
3
通讯作者：
J. Beale
中科院分区：
文献类型：
--
作者：
J. Beale
本文证明了以略大于临界速度传播的孤立水波的存在性。所使用的方法是基于纳什-莫泽隐函数定理的一个版本。它比以前的存在性证明要简单一些，而且还显示了波对决定速度和振幅的参数的平滑依赖。这证明用参数的幂展开法来计算波是合理的。所谓孤立水波，我们指的是二维无旋流，这样流体在无穷远处静止，相对于匀速运动的坐标系，流动是稳定的。我们假设底部是水平的，流体运动是由压力和重力决定的。近似计算表明，孤立波的传播速度U总是大于临界速度（gh） ‘ ’，其中h是流体在无穷远处的深度，g是重力加速度。这种波的存在的第一个证明是由拉夫连捷耶夫获得的，他认为当波长增加到无穷大时，它们是周期波的极限。Friedrichs和Hyers给出了一个更直接的证明。在物理坐标中，这个问题涉及到自由边界，很难直接处理。出于这个原因，我们在这里采用了一个由列维-奇维塔引入的问题的转换，并在b[2]中使用。关于这种转换的详细讨论由Stoker在b[7]，第10.9和12.2节中给出；我们在这里只描述问题的结果形式。如果速度场（u, u）接近于恒定水平流，则w= u-iu在条带=((4,$):O< $< l}中作为4+ i $的解析函数就足够求解，其中4为速度势，$为流函数；所有变量都是非量纲化的。设w= exp {-i (O'+ iA')}，则边界条件为
In this paper we present a proof of the existence of solitary water waves traveling at velocities slightly greater than the critical velocity. The method used is based on a version of the Nash-Moser Implicit Function Theorem. It is somewhat simpler than earlier existence proofs and shows in addition the smooth dependence of the wave on a parameter determining the velocity and amplitude. This justifies a calculation of the wave by expansion in powers of the parameter.By a solitary water wave we mean a two-dimensional irrotational flow such that the fluid is at rest at infinity and the flow is steady with respect to a coordinate system moving at a constant velocity. We assume that the bottom is horizontal and the fluid motion is determined by pressure forces and gravity. Approximate calculations indicate that the propagation velocity U of the solitary wave is always greater than the critical velocity (gh)"', where h is the depth of the fluid at infinity and g is the acceleration of gravity. The first proof of the existence of such waves was obtained by Lavrentiev, who considered them as limits of periodic waves, as the wavelength increased to infinity. A more direct proof was given by Friedrichs and Hyers [2]. In physical coordinates the problem involves a free boundary and is hard to deal with directly. For this reason we employ here a transformation of the problem introduced by Levi-Civita and used in [2]. A detailed discussion of this transformation is given by Stoker in [7], Sections 10.9 and 12.2 a; we merely describe the resulting form of the problem here. If the velocity field (u, u) is close to a constant horizontal flow, it is sufficient to solve for w= u-iu as an analytic function of 4+ i $ in the strip=((4, $): O< $< l}, where 4 is the velocity potential and $ is the stream function; all variables have been nondimensionalized. If we set w= exp {-i (O'+ iA')} the boundary conditions are