Anatoly Kistovich

The theoretical study of steady vortex motion of homogeneous ideal heavy fluid with a free surface by methods of differential geometry is presented. The main idea of methods is based on suggestion that a velocity field is formed by geodesic flows at some surfaces. For steady flow integral flow lines are geodesics on the second order surfaces being parameterized and located in the space occupied by the fluid. In this case both Euler and continuity equations are transformed into equations for inner geometry parameters. Conditions on external parameters are derived from boundary conditions of the problem. The investigation of properties of generalized Rankine vortex that is vertical vortex flow contacting with a free surface is done. In supplement to the classical Rankine vortex these vortices are characterized by all finite specific integral invariants. The constructed set of explicit solutions depend on a unique parameter, which can be defined experimentally through measurements of depth and shape of a near surface hole produced by the vortex.

A new constructive regular method to search discrete symmetries of a set of differential equations is proposed. The method is based on automorphism properties of a basic $1$-forms for initial and transformed equations. In difference with well-known methods of discrete symmetry searching the proposed method does not need in a priori knowledge and as sequence in a preliminary searching of infinitesimal symmetries for studied equations.