Kink solutions of the binormal flow

Luis Vega

Journées équations aux dérivées partielles (2003)

  • page 1-10
  • ISSN: 0752-0360

Abstract

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I shall present some recent work in collaboration with S. Gutierrez on the characterization of all selfsimilar solutions of the binormal flow : X t = X s × X s s which preserve the length parametrization. Above X ( s , t ) is a curve in ℝ 3 , s ∈ ℝ the arclength parameter, and t denote the temporal variable. This flow appeared for the first time in the work of Da Rios (1906) as a crude approximation to the evolution of a vortex filament under Euler equation, and it is intimately related to the focusing cubic nonlinear Schrödinger equation through the so called Hasimoto transformation. These solutions show the formation of singularities in finite time in the shape of either just a kink (zero angular momentum) or a kink together with a logarithmic correction in the shape of a spiral (non trivial angular momentum).

How to cite

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Vega, Luis. "Kink solutions of the binormal flow." Journées équations aux dérivées partielles (2003): 1-10. <http://eudml.org/doc/93441>.

@article{Vega2003,
abstract = {I shall present some recent work in collaboration with S. Gutierrez on the characterization of all selfsimilar solutions of the binormal flow : $X_\{t\}=X_\{s\}\times X_\{ss\}$ which preserve the length parametrization. Above $X(s,t)$ is a curve in $\mathbb \{R\}^3$, $s\in \mathbb \{R\}$ the arclength parameter, and $t$ denote the temporal variable. This flow appeared for the first time in the work of Da Rios (1906) as a crude approximation to the evolution of a vortex filament under Euler equation, and it is intimately related to the focusing cubic nonlinear Schrödinger equation through the so called Hasimoto transformation. These solutions show the formation of singularities in finite time in the shape of either just a kink (zero angular momentum) or a kink together with a logarithmic correction in the shape of a spiral (non trivial angular momentum).},
author = {Vega, Luis},
journal = {Journées équations aux dérivées partielles},
keywords = {selfsimilar solutions; binormal flow; vortex filament; vorticity; Biot-Savart integral},
language = {eng},
pages = {1-10},
publisher = {Université de Nantes},
title = {Kink solutions of the binormal flow},
url = {http://eudml.org/doc/93441},
year = {2003},
}

TY - JOUR
AU - Vega, Luis
TI - Kink solutions of the binormal flow
JO - Journées équations aux dérivées partielles
PY - 2003
PB - Université de Nantes
SP - 1
EP - 10
AB - I shall present some recent work in collaboration with S. Gutierrez on the characterization of all selfsimilar solutions of the binormal flow : $X_{t}=X_{s}\times X_{ss}$ which preserve the length parametrization. Above $X(s,t)$ is a curve in $\mathbb {R}^3$, $s\in \mathbb {R}$ the arclength parameter, and $t$ denote the temporal variable. This flow appeared for the first time in the work of Da Rios (1906) as a crude approximation to the evolution of a vortex filament under Euler equation, and it is intimately related to the focusing cubic nonlinear Schrödinger equation through the so called Hasimoto transformation. These solutions show the formation of singularities in finite time in the shape of either just a kink (zero angular momentum) or a kink together with a logarithmic correction in the shape of a spiral (non trivial angular momentum).
LA - eng
KW - selfsimilar solutions; binormal flow; vortex filament; vorticity; Biot-Savart integral
UR - http://eudml.org/doc/93441
ER -

References

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  1. [Be] R. Betchov, On the curvature and torsion of an isolated filament, Journal of Fluids Mechanics 22, (1965), 471. Zbl0133.43803MR178656
  2. [Bu] T. F. Buttke, A numerical study of superfluid turbulence in the Self-Induction Approximation, J. of Com. Phys. 76, (1988), 301-326. Zbl0639.76136
  3. [CFM] P. Constantin, C. Fefferman, A. Majda, Geometric constraints on potentially singular solutions for 3-D Euler equations, Comm. P.D.E. 21, (1988), 559-571. Zbl0853.35091MR1387460
  4. [DaR] L.S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo 22, (1906), 117. JFM37.0764.01
  5. [GRV] S. Gutiérrez, J. Rivas, L. Vega, Formation of singularities and self-similar vortex motion under the Localized Induction Approximation, To appear in Comm. P.D.E. . Zbl1044.35089MR1986056
  6. [Ha] H. Hasimoto, A soliton on a vortex filament, Journal of Fluids Mechanics, 51, (1972), 477-485. Zbl0237.76010
  7. [LRT] M. Lakshmanan, T.H. W. Ruijgrok, C.J. Thompson, On the dynamics of a continuum spin system, Physica A 84, (1976), 577-590. MR449262
  8. [Sa] P.G. Saffman, Vortex Dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge Univ. Press, New York, (1992). Zbl0777.76004MR1217252

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