Nonlinear waves, computer algebra and vortex dynamics

Abstract

New solutions of two dimensional, inviscid, steady vortex dynamics are derived by techniques from the theory of solitons and nonlinear waves. The case where the vorticity and the stream function are related by the hyperbolic sinh function serves as an illustrative example. The ‘positon’ of certain nonlinear evolution equations is obtained by a special coalescence of wavenumbers in the multi-soliton solution. The ‘positon’ of the sinh-Poisson equation is nonsingular, and the streamlines consist of a sequence of tripoles in the long wave limit. Computer algebra software is employed to verify the validity of the solutions independently. Relevance of these novel solutions and comparison with similar works in the literature are discussed.