Solutions of the solitary wave type on the surface of an infinite depth fluid with a submerged point vortex

In the paper, we consider a two-dimensional free-surface flow past a single point vortex in fluid of infinite depth. The flow moves from left to right with uniform speed $c$ far upstream and is subject to the downward acceleration $g$ of gravity. A point vortex of circulation $\varGamma$ is located at depth $H$ . The positive direction of circulation is counterclockwise. The flow is characterised by two dimensionless parameters which are the dimensionless vortex circulation $\gamma =\varGamma /(\textit{cH}\,)$ and the Froude number $ \textit{Fr}=c/\sqrt {gH}$ . The goal of the paper is to find the solutions of the solitary wave type with one or several crests on the free surface. These solutions are waveless far downstream and have a vertical line of symmetry. We have established that for a fixed Froude number $ \textit{Fr}\le 0.8$ , there exists a finite set of positive $\gamma$ for which the solutions of the solitary wave type occur.