On a vortex filament with the axial velocity

Vortex filament, as one of the fundamental elements underlying the dynamics in high-Reynolds-number flows, has garnered researchers’ attention. Motion of the nonlinear waves on a thin vortex filament with axial velocity, embedded in an inviscid incompressible fluid, is investigated in this paper. Due to the second-order effect (SOE), the curvature and torsion of the vortex filament are governed by a disturbed Fukumoto–Miyazaki equation. Bilinear forms and nonlinear waves including the bright, multipole and double-pole solitons, are constructed. Solutions for the disturbed Fukumoto–Miyazaki equation present that: The SOE does not change the paths and shapes of the solitons on the vortex filament, while it changes the velocities of those solitons; Multipole solitons propagate around the axis periodically and their shapes change periodically, while when the SOE approaches to 0, the period of the shape change approaches to infinity, and the multipole solitons can not exist; Dipole-like solitons exist unstable and they will degenerate into two solitons because of the SOE; Breathers can also exist on the helicoidal vortex filaments, and the SOE changes the velocities of the breathers, while the amplitudes of the breathers are not affected by the SOE; Rogue waves appear suddenly on the filaments and their amplitudes reach the peaks and troughs simultaneously; Because of the SOE, the appearances of the rogue waves are accompanied by the appearances of the slight shocks.