Self-stretching of perturbed vortex filaments. II. Structure of solutions

Recently, the authors have derived a new asymptotic equation for the evolution of small amplitude, short wavelength perturbations of slender vortex filaments which differs significantly from the familiar self-induction approximation. One important difference is that the equation includes some of the effects of vortex stretching in a simple fashion. Through a filament function, this new asymptotic equation becomes a cubic nonlinear Schrödinger equation perturbed by an explicit nonlocal operator. In this paper, several prominent features of solutions of this asymptotic equation are analyzed in detail through a combination of mathematical analysis, exact solutions, and numerical computations. The main result of the analysis is that the nonlocal operator generates a novel and remarkable singular perturbation of the cubic nonlinear Schrödinger equation involving a strongly indefinite Hamiltonian structure. In the numerical calculations reported here, the filament function develops higher and much narrower peaks as time evolves when compared with the corresponding solutions of NLS. Furthermore, in many of the examples, we demonstrate that these curvature peaks correspond to the birth of small scale "hairpins" or kinks along the actual vortex filament. Thus, the new asymptotic equation yields the birth of short wavelength hairpins along a perturbed vortex filament.