Evolution of polygonal lines by the binormal flow

TL;DR

This paper explores the binormal flow related to vortex filament dynamics, linking it to the nonlinear Schrödinger equation, demonstrating phenomena like the Talbot effect, and establishing existence and continuation of solutions with polygonal initial data.

Contribution

It establishes the existence and uniqueness of solutions to the binormal flow with polygonal initial data and connects these solutions to the nonlinear Schrödinger equation, including phenomena like intermittency and continuation after blow-up.

Findings

Solutions of the cubic nonlinear Schrödinger equation linked to Dirac mass initial data.

Observation of the Talbot effect in the context of the equation.

Existence and continuation of binormal flow solutions with polygonal lines.

Abstract

The aim of this paper is threefold. First we display solutions of the cubic nonlinear Schr{\"o}dinger 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 Schr{\"o}dinger equation.