Abstract
We study the linear stability of small-amplitude torus knot solutions of the localized induction approximation equation for the motion of a thin vortex filament in an ideal fluid. Such solutions can be constructed analytically through the connection with the focusing nonlinear Schrödinger equation using the method of isoperiodic deformations. We show that these (p, q) torus knots are generically linearly unstable for p < q, while we provide examples of neutrally stable (p, q) torus knots with p > q, in contrast with an earlier linear stability study by Ricca (1993 Chaos 3 83–95; 1995 Chaos 5 346; 1995 Small-scale Structures in Three-dimensional Hydro and Magneto-dynamics Turbulence (Lecture Notes in Physics vol 462) (Berlin: Springer)). We also provide an interpretation of the original perturbative calculation in Ricca (1995), and an explanation of the numerical experiments performed by Ricca et al (1999 J. Fluid Mech. 391 29–44), in light of our results.
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