Soliton propagation on vortex cores and the Hasimoto soliton

Weakly nonlinear, nonaxisymmetric wave propagation on vortex cores is considered under the assumption of inviscid, incompressible flow. The particular vortex ‘‘waveguide’’ upon which the waves are assumed to propagate has a vorticity distribution typical of experimentally realized concentrated vortices. The waves are modulated, and the modulation envelope is governed by the nonlinear Schrödinger equation. Solitons are found to be possible in a restricted band of wavenumbers having both a high and low wavenumber cutoff. It is found that the vortex deformations resulting from this analysis can be matched with Hasimoto’s soliton, which is based upon a local induction approximation and therefore ignores the vorticity dynamics in the core. This allows one to determine the ratio of Biot–Savart cutoffs which appears as an undetermined constant in Hasimoto’s theory: in effect, this calibrates the Hasimoto soliton. In addition, the present analysis shows that the group velocity of solitons with deflections small compared to the core radius is at most 0.292 Γ/2πa, where Γ is the circulation of the vortex, and a is the core radius; this may explain recent experimental observations of Hopfinger, Browand, and Gagne [J. Fluid Mech. 125, 505 (1982)].