Evolution of Polygonal Lines by the Binormal Flow

• Published in: Annals of PDE Vol. 6; no. 1; p. 6
• Main Authors: Banica, Valeria, Vega, Luis
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-06-2020; Springer Nature B.V; Springer
• Subjects: Analysis of PDEs; Cauchy problems; Filaments; Mathematical Methods in Physics; Mathematics; Partial Differential Equations; Physics; Physics and Astronomy; Polygons; Reynolds number; Schrodinger equation; Talbot effect; Vortex filaments; Vortices; Talbot effect; Nonlinear Schrödinger equations; Binormal flow; Singular data; Vortex filaments
• ISSN: 2524-5317; 2199-2576
• DOI: 10.1007/s40818-020-0078-z

The aim of this paper is threefold. First we display solutions of the cubic nonlinear Schrödinger equation on
R
in link with initial data a sum of Dirac masses. Secondly we show a Talbot effect for the same equation. Finally we prove the existence of a unique solution of the binormal flow with datum a polygonal line. This equation is used as a model for the vortex filaments dynamics in 3-D fluids and superfluids. We also construct solutions of the binormal flow that present an intermittency phenomena. Finally, the solution we construct for the binormal flow is continued for negative times, yielding a geometric way to approach the continuation after blow-up for the 1-D cubic nonlinear Schrödinger equation.