Solitons, Euler’s equation, and vortex patch dynamics

Integrable systems related to the Korteweg--de Vries (KdV) equation are shown to be associated with the dynamics of vortex patches in ideal two-dimensional fluids. This connection is based on a truncation of the exact contour dynamics analogous to the ``localized induction approximation'' which relates the nonlinear Schr\"odinger equation to the motion of a vortex filament. Single soliton solutions of the periodic modified KdV problem correspond to uniformly rotating shapes which approximate the Kirchoff ellipse and known generalizations. A simple geometrical interpretation of the dual Poisson bracket structure of the modified KdV hierarchies is given.