Short branch cut approximation in two-dimensional hydrodynamics with free surface

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