Formation of singularities on the free surface of an ideal fluid

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Abstract
It is shown that the equations of motion of an ideal fluid with a free surface in the absence of both gravitational and capillary forces can be effectively solved in the approximation of small surface angles. It can be done by means of an analytical continuation of both the velocity potential on the surface and its elevation. For almost arbitrary initial conditions the system evolves to the formation of singularities in a finite time. Three kinds of singularities are shown to be possible. The first one is of the root character provided by the analytical behavior of the velocity potential. In this case the process of the singularity formation, representing some analog of the wave breaking, is described as a motion of branch points in the complex plane towards the real axis. The second type can be obtained as a result of the interaction of two movable branch points leading to the formation of wedges on the free surface. The third kind is associated with a motion in the complex plane of the singular points of the analytical continuation of the elevation, resulting in the appearance of strong singularities for the surface profile.

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