Branch cuts of Stokes wave on deep water. Part II: Structure and location of branch points in infinite set of sheets of Riemann surface
Branch cuts of Stokes wave on deep water. Part II: Structure and location of branch points in infinite set of sheets of Riemann surface
复制标题
深水中斯托克斯波浪的树枝切割。
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Pavel M. Lushnikovy
中科院分区：
文献类型：
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作者：
Pavel M. Lushnikovy
将Stokes波的UID面映射到实线，将UID域映射到下复半平面。斯托克斯的每个空间周期有一个平方根分支点，位于距实轴vc的上复半平面上。随着Stokes波高的增加，Vc趋近于零，在Vc=0时形成极限Stokes波：极限Stokes波具有2=3次幂奇点，在波峰上形成2=3弧度角，这与对任意小但非零的Vc有效的平方根奇点有本质的不同，使得零Vc的极限非常重要。这一极限是通过将平方根的分支切割交叉到黎曼曲面的第二个和更高的薄片中，以在每个薄片与实轴的距离Vc处找到耦合的平方根奇点来解决的。薄片的数量是固有的，并且可以找到斯托克斯波在所有这些薄片中的解析延拓，以及每个薄片内VC奇点的半整数次级数展开。推测导数阶的非极限斯托克斯波由嵌套平方根奇点的固有数组成。当VC消失时，这些嵌套的平方根形成极限斯托克斯波的2=3幂定律奇点。
uid surface of Stokes wave into the real line with uid domain mapped into the lower complex half-plane. There is one square root branch point per spatial period of Stokes located in the upper complex half-plane at the distance vc from the real axis. The increase of Stokes wave height results in approaching vc to zero with the limiting Stokes wave formation at vc = 0: The limiting Stokes wave has 2=3 power law singularity forming 2 =3 radians angle on the crest which is qualitatively different from the square root singularity valid for arbitrary small but nonzero vc making the limit of zero vc highly nontrivial. That limit is addressed by crossing a branch cut of square root into the second and higher sheets of Riemann surface to nd coupled square root singularities at the distances vc from the real axis at each sheet. The number of sheets is innite and the analytical continuation of Stokes wave into all these sheets is found together with the series expansion in half-integer powers at vc singular points within each sheet. It is conjectured that non-limiting Stokes wave at the leading order consists of the innite number of nested square root singularities. These nested square roots form 2=3 power law singularity of the limiting Stokes wave as vc vanishes.