A potential motion of ideal incompressible fluid with a free surface and infinite depth is considered in two-dimensional geometry. A time-dependent conformal mapping of the lower complex half-plane of the auxiliary complex variable w into the area filled with fluid is performed with the real line of w mapped into the free fluid’s surface. The fluid dynamics can be fully characterized by the motion of the complex singularities in the analytical continuation of both the conformal mapping and the complex velocity. We consider the short branch cut approximation of the dynamics with the small parameter being the ratio of the length of the branch cut to the distance between its centre and the real line of w. We found that the fluid dynamics in that approximation is reduced to the complex Hopf equation for the complex velocity coupled with the complex transport equation for the conformal mapping. These equations are fully integrable by characteristics producing the infinite family of solutions, including moving square root branch points and poles. These solutions involve practical initial conditions resulting in jets and overturning waves. The solutions are compared with the simulations of the fully nonlinear Eulerian dynamics giving excellent agreement even when the small parameter approaches about one.

在二维几何结构中考虑了具有自由表面和无限深度的理想不可压缩流体的潜在运动。辅助复数变量w的下复数半平面到充满流体的区域的时间依赖共形映射是通过将w的实线映射到自由流体的表面中来执行的。在共形映射和复数速度的解析连续过程中，复数奇异点的运动可以完全表征流体动力学。我们考虑动力学的短分支切分近似，其中小参数是分支切分的长度与其中心与w的实线之间的距离的比值。我们发现，在这种近似中，流体动力学被简化为复数速度的复数霍普夫方程和共形映射的复数输运方程。这些方程式具有产生无限解决方案族的特征，包括移动的平方根分支点和极点，因此可以完全积分。这些解决方案涉及实际的初始条件，从而导致喷射和倾覆波。将该解决方案与全非线性欧拉动力学的仿真进行了比较，即使当小参数接近1时，也具有极好的一致性。