High-Jacobian approximation in the free surface dynamics of an ideal fluid

Using a combination of the canonical formalism for free-surface hydrodynamics and conformal mapping to a horizontal strip we obtain a simple system of pseudo-differential equations for the surface shape and hydrodynamic velocity potential, The system is well-suited for numerical simulation. It can be effectively studied in the case when the Jacobian of the conformal mapping takes very high values in the vicinity of some point on the surface. At first order in an expansion in inverse powers of the Jacobian one can reduce the whole system of equations to a single equation which coincides with the well-known Laplacian Growth Equation (LGE). In the framework of this model one can construct remarkable special solutions of the system describing such physical phenomena as formation of finger-type configurations or changing of the surface topology generation of separate droplets.