Algorithm for constructing exact solutions of the problem of unsteady plane motion of a fluid with a free surface

Unsteady plane potential flows of an ideal incompressible fluid with a free surface in the absence of external forces and capillarity are studied. An algorithm to construct exact solutions for such flows has been proposed on the basis of the analysis of compatibility conditions of the equations of motion and an auxiliary complex transport equation. This algorithm makes it possible to significantly expand the list of known exact nontrivial solutions of the considered classical problem. This list has recently consisted of only a few Dirichlet solutions: flows for which the surface of the fluid is a parabola, ellipse, or hyperbola. This algorithm allows reproducing a recently found class of solutions specified by the Hopf equation for the complex velocity and finding a fundamentally new large class of solutions for which flows are described by the Hopf equation for the inverse complex velocity.