Knotted vortex filaments in an ideal fluid

Knotted closed-curve solutions of the equation of self-induced vortex motion are studied. It is shown that there are invariant torus knots which translate and rotate as rigid bodies. The general motion of ‘small-amplitude’ torus knots and iterated (cabled) torus knots is described and found to be almost periodic in time, and for some, but not all, initial data, the topology of the knot is shown to be invariant.

[1]  A. Callegari,et al.  Motion of a curved vortex filament with decaying vortical core and axial velocity , 1978 .

[2]  H. K. Moffatt,et al.  The degree of knottedness of tangled vortex lines , 1969, Journal of Fluid Mechanics.

[3]  Schwarz.  Three-dimensional vortex dynamics in superfluid 4He: Homogeneous superfluid turbulence. , 1988, Physical review. B, Condensed matter.

[4]  Stephen Omohundro,et al.  ON THE GLOBAL STRUCTURE OF PERIOD DOUBLING FLOWS , 1984 .

[5]  P. Holmes,et al.  Nonlinear oscillators, iterated maps, symbolic dynamics, and knotted orbits , 1987, Proceedings of the IEEE.

[6]  Joan S. Birman,et al.  Knotted periodic orbits in dynamical systems—I: Lorenz's equation , 1983 .

[7]  S. Kida.  A vortex filament moving without change of form , 1981, Journal of Fluid Mechanics.

[8]  James P. Keener,et al.  Solitons under perturbations , 1977 .

[9]  H. Hasimoto,et al.  A soliton on a vortex filament , 1972, Journal of Fluid Mechanics.

[10]  Chen,et al.  Nonlinear self-modulation: An exactly solvable model. , 1988, Physical review. A, General physics.

[11]  P. Holmes.  Knotted periodic orbits in suspensions of annulus maps , 1987, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[12]  H. K. Moffatt.  Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 1. Fundamentals , 1985, Journal of Fluid Mechanics.

[13]  Philip Holmes,et al.  Knotted periodic orbits in suspensions of Smale's horseshoe: Torus knots and bifurcation sequences , 1985 .

[14]  Joel Langer,et al.  Knotted Elastic Curves in R3 , 1984 .

[15]  J. Keener.  Knotted scroll wave filaments in excitable media , 1989 .

[16]  S. Kida.  Stability of a Steady Vortex Filament , 1982 .

[17]  Schwarz.  Three-dimensional vortex dynamics in superfluid 4He: Line-line and line-boundary interactions. , 1985, Physical review. B, Condensed matter.

[18]  Joseph B. Keller,et al.  Ropes in equilibrium , 1987 .

[19]  James P. Keener,et al.  The dynamics of three-dimensional scroll waves in excitable media , 1988 .

[20]  J. Keener,et al.  The Motion of Untwisted Untorted Scroll Waves in Belousov-Zhabotinsky Reagent , 1988, Science.

[21]  R. Betchov.  On the curvature and torsion of an isolated vortex filament , 1964, Journal of Fluid Mechanics.