Solitons and 2 D vortex dynamics

Abstract

In inviscid, steady, two dimensional flows without body force, one general solution of the equations of motion is ømega = f(\psi), where ømega = vorticity, \psi = stream function, f = a differentiable but otherwise arbitrary function. Recent advances in the theory of solitons and nonlinear waves will be employed to obtain new solutions in vortex dynamics. More precisely, the sine - Gordon and the sinh - Poisson equations will be treated. Known solutions in the literature can be re-derived. Both localized solitons and periodic waves will be utilized. New solutions include for example a doubly periodic array of vortices and a vortex in a box. The new concept of a `positon' will also be examined.