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Linear and Non-Linear Waves

We use the so-called localized Induction Approximation as a mathematical model to describe the evolution of vortex filaments. Thanks to the Hasimoto transformation this approximation is related to the cubic-non linear Schrödinger equation. According to this model, self similar solutions are very easy to characterize and, at least qualitatively, their shapes are very close to those exhibited by the smoke from a cigarette and by the vortices generated behind a triangular wing.

We also study the evolution of filaments that at an initial time are given by a regular polygon. In that case the dynamics is determined by the well known Gauss Sums and therefore it heavily depends on how "rational" is the time that is considered. More concretely at a rational time t=p/q the filament is a skew-polygon with a number of sides that is proportional to q. Moreover the trajectory seems to be periodic and the starting polygon is reproduced at half of a period but with the axes switched. For irrational times a fractal appears.

Selfsimilar solutions versus vortices behind a triangular wind

Curlicue in the unit sphere generated at an irrational time

Eusko Jaurlaritza - Gobierno Vasco ikerbasque - Basque Foundation for Science Bizkaia xede. Bizkaiko Foru Aldundia innobasque - Agencia vasca de la innovación Universidad del PaÌs Vasco (UPV/EHU)