Three-dimensional distortions of a vortex filament: Exact solutions of the localized induction equation

The three-dimensional motion of a thin vortex filament in an inviscid incompressible fluid is investigated theoretically, on the basis of the localized induction equation(LIE). It is shown that the N-soliton solution, obtained through Hirota's bilinear method, does not exhibit clear phase-advance during head-on collisions as observed in the experiment by Maxworthy et al. In order to resolve this discrepancy an effect of axial flow within the vortex core is incorporated into the LIE and a new integrable equation is derived. The bilinear procedure as well as the soliton surface approach gives the N-soliton solution which is identical to that of the LIE except for the dispersion relation. Besides, this equation predicts that a certain class of helicoidal vortices with axial flow is neutrally stable against any small perturbations.