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A soliton on a vortex filament

  • Hidenori Hasimoto (a1)

The intrinsic equation governing the curvature K and the torsion τ of an isolated very thin vortex filament without stretching in an incompressible inviscid fluid is reduced to a non-linear Schrödinger equation \[ \frac{{\rm l}}{i}\frac{\partial \psi}{\partial t} = \frac{\partial^2\psi}{\partial s^2}+{\textstyle\frac{1}{2}}(|\psi|^2+A)\psi, \] where t is the time, s the length measured along the filament, ψ is the complex variable \[ \psi = \kappa\exp\left(i\int_0^{s}\tau \,ds\right) \] and is a function oft. It is found that this equation yields a solution describing the propagation of a loop or a hump of helical motion along a line vortex, with a constant velocity 2τ. The relation to the system of intrinsic equations derived by Betchov (1965) is discussed.

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Asano N., Taniuti, T. & Yajima N.1969 J. Math. Phys. 10, 2020.
Batchelor G. K.1967 An Introduction to Fluid Dynamics, p. 509. Cambridge University Press.
Betcrov R.1965 J. Fluid Mech. 22, 471.
Hama F. R.1962 Phys. Fluids, 5, 1156.
Hama F. R.1963 Phys. Fluids, 6, 526.
Hasimoto H.1971 J. Phys. Soc. Japan, 31, 293.
Kambe, T. & Takao T.1971 J. Phys. Soc. Japan, 31. 591.
Karpman, V. I. & Krushkal E. M.1969 Soc. Phys. J.E.T.P., 28, 277.
Taniuti, T. & Yajima N.1969 J. Math. Phys. 10, 1369.
Yajima, N. & Outi A.1971 Prog. Theor. Phys. (Kyoto) 45, 1997.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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