Dynamics of poles in two-dimensional hydrodynamics with free surface: new constants of motion
Dynamics of poles in two-dimensional hydrodynamics with free surface: new constants of motion
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DOI：
10.1017/jfm.2019.448
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发表时间：
2018-09
影响因子：
3.7
通讯作者：
A. Dyachenko;S. Dyachenko;P. Lushnikov;Vladimir E Zakharov
中科院分区：
文献类型：
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作者：
A. Dyachenko;S. Dyachenko;P. Lushnikov;Vladimir E Zakharov
我们研究了二维几何中具有自由表面和无限深度的理想不可压缩流体的势运动问题。我们承认重力和表面张力的存在。将变量$w$的下复半平面$z(w,t)$映射到充满流体的区域，并将$w$的实线映射到自由流体的表面。研究了$z(w,t)$和$\unicode[STIX]{x1D6F1}(w,t)$在$w$的上复半平面上的奇异性动力学。我们证明了$z_{w}(w,t)$和$\unicode[STIX]{x1D6F1}_{w}(w,t)$中复极点的任意有限数$N$解的存在性，它们是$z(w,t)$和$\unicode[STIX]{x1D6F1}(w,t)$的导数。我们强调这些解不是纯有理的，因为它们通常在复半平面上的其他位置有分支点。当表面张力为零时，极点的阶数可以是任意的，而当表面张力为非零时，极点的阶数都是偶数。我们发现$z_{w}(w,t)$在这$N$点处的残数是新的，以前未知的运动常数，参见Zakharov & Dyachenko（2012，作者未发表的观测，arXiv:1206.2046）的初步结果。所有这些运动常数在潜在的哈密顿动力学意义上相互交换。在没有重力和表面张力的情况下，$\unicode[STIX]{x1D6F1}_{w}(w,t)$的残量也是运动常数，而非零重力$g$保证了这些残量对时间的线性依赖。$z_{w}(w,t)$和$\unicode[STIX]{x1D6F1}_{w}(w,t)$在每个极点位置的Laurent级数展开式揭示了二阶极点的附加运动积分的存在。如果所有的极点都是简单的，那么运动的独立实积分的数量在零重力下是4N，在非零重力下是4N-1。对于二阶极点，我们发现零重力下的运动积分为6N，非零重力下的运动积分为6N-1。我们认为这些非平凡运动常数的存在为深水自由表面流体力学的完全可积猜想提供了一个论据。分析结果得到了高精度数值的有力支持。
We address the problem of the potential motion of an ideal incompressible fluid with a free surface and infinite depth in a two-dimensional geometry. We admit the presence of gravity forces and surface tension. A time-dependent conformal mapping $z(w,t)$ of the lower complex half-plane of the variable $w$ into the area filled with fluid is performed with the real line of $w$ mapped into the free fluid’s surface. We study the dynamics of singularities of both $z(w,t)$ and the complex fluid potential $\unicode[STIX]{x1D6F1}(w,t)$ in the upper complex half-plane of $w$ . We show the existence of solutions with an arbitrary finite number $N$ of complex poles in $z_{w}(w,t)$ and $\unicode[STIX]{x1D6F1}_{w}(w,t)$ which are the derivatives of $z(w,t)$ and $\unicode[STIX]{x1D6F1}(w,t)$ over $w$ . We stress that these solutions are not purely rational because they generally have branch points at other positions of the upper complex half-plane. The orders of poles can be arbitrary for zero surface tension while all orders are even for non-zero surface tension. We find that the residues of $z_{w}(w,t)$ at these $N$ points are new, previously unknown, constants of motion, see also Zakharov & Dyachenko (2012, authors’ unpublished observations, arXiv:1206.2046) for the preliminary results. All these constants of motion commute with each other in the sense of the underlying Hamiltonian dynamics. In the absence of both gravity and surface tension, the residues of $\unicode[STIX]{x1D6F1}_{w}(w,t)$ are also the constants of motion while non-zero gravity $g$ ensures a trivial linear dependence of these residues on time. A Laurent series expansion of both $z_{w}(w,t)$ and $\unicode[STIX]{x1D6F1}_{w}(w,t)$ at each poles position reveals the existence of additional integrals of motion for poles of the second order. If all poles are simple then the number of independent real integrals of motion is $4N$ for zero gravity and $4N-1$ for non-zero gravity. For the second-order poles we found $6N$ motion integrals for zero gravity and $6N-1$ for non-zero gravity. We suggest that the existence of these non-trivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics in deep water. Analytical results are solidly supported by high precision numerics.