Kink solutions of the binormal flow
Journées équations aux dérivées partielles (2003), article no. 14, 10 p.

I shall present some recent work in collaboration with S. Gutierrez on the characterization of all selfsimilar solutions of the binormal flow : ${X}_{t}={X}_{s}×{X}_{ss}$ which preserve the length parametrization. Above $X\left(s,t\right)$ is a curve in ${ℝ}^{3}$, $s\in ℝ$ 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).

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author = {Vega, Luis},
title = {Kink solutions of the binormal flow},
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publisher = {Universit\'e de Nantes},
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Vega, Luis. Kink solutions of the binormal flow. Journées équations aux dérivées partielles (2003), article  no. 14, 10 p. doi : 10.5802/jedp.628. http://archive.numdam.org/articles/10.5802/jedp.628/

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