N-Solitons on a Curved Vortex Filament with Axial Flow

The long bending motion of a thin vortex filament with axial flow inside the core is investigated theoretically. The fluid outside the vortex core is assumed to be incompressible, inviscid and irrotational. A new integrable nonlinear equation, valid under the assumption of local induction, is proposed. This equation is shown to be equivalent to the Hirota equation and the N-soliton solution is given explicitly, using the Hirota's method of bilinear form.