Nonlinear dynamics of the free surface of an ideal fluid

On the basis of a combination of the canonical formalism for describing free-surface fluid of finite depth and conformal mapping to a horizontal strip, a simple system of pseudo-differential equations is obtained for the surface shape η and the hydrodynamic velocity potential Ψ. This system can be effectively studied in the case of small-angle deviations of the fluid surface from a plane surface (without gravity and surface tension) and, in the opposite case, when the value of the Jacobian of the conformai mapping is very large in the vicinity of some point on the surface. In the first case, the complex velocity potential obeys a Hopf-type differential equation. In this approximation, singularities on the surface appear as a result of violation of the analyticity of the potential in the strip due to the motion of singularities (i.e., the branch points) towards the strip boundaries. These singularities are shown to be mainly of the root type. For such singularities, the curvature becomes infinite, η_xx∼|x|^-1/2, but the surface remains smooth. In the other limiting case, in a first-order expansion in inverse powers of the Jacobian, the whole system of equations can be reduced to a single equation that coincides with the well-known Laplacian Growth Equation. Within this model, remarkable special solutions of the system can be constructed describing such physical phenomena as the formation of finger-type configurations or the generation of separate droplets changing the surface topology.