On vorticity-induced waves in a homogeneous incompressible fluid (Q1070925)

The existence of vortex-induced waves in a homogeneous incompressible fluid is proved. The boundary of the vortex represents a cylindrical rotating fluid surface of stable form. The nonlinear dispersion relation, the form of the vortex and the stream function are found for the vortices bounded by an almost circular cylinder, and for the vortex-induced waves. The character and special features of the oscillation of the velocity field are explained. The problem is reduced to that of proving the existence of a branching solution of the nonlinear integral equation and to effective determination of the solution and the bifurcation value of the parameter. An iterative method is proposed enabling the simultaneous determination at every stage of the approximation to the branching solution and bifurcation value of the parameter. The convergence of the method over a certain range of parameters is proved.