Turbulent solutions of the binormal flow and the 1D cubic Schr\"odinger equation

TL;DR

This paper explores turbulent phenomena in the 1D cubic Schr"odinger equation and the binormal flow, revealing complex behaviors like singularities and multifractality, and constructing well-posedness in critical function spaces.

Contribution

It rigorously analyzes turbulent behaviors in these equations, including singularity formation and Fourier growth, and establishes well-posedness in critical Fourier-Lebesgue and supercritical Sobolev spaces.

Findings

Demonstrated creation of singularities and Fourier growth.

Established well-posedness in critical Fourier-Lebesgue space.

Connected vortex filament dynamics to turbulence phenomena.

Abstract

In the last three decades there has been an intense activity on the exploration of turbulent phenomena of dispersive equations, as for instance the growth of Sobolev norms since the work of Bourgain in the 90s. In general the 1D cubic Schr\"odinger equation has been left aside because of its complete integrability. In a series of papers of the last six years that we survey here for the special issue of the ICMP 2024 ([12],[13],[14],[15],[16],[7],[8]), we considered, together with the 1D cubic Schr\"odinger equation, the binormal flow, which is a geometric flow explicitly related to it. We displayed rigorously a large range of complex behavior as creation of singularities and unique continuation, Fourier growth, Talbot effects, intermittency and multifractality, justifying in particular some previous numerical observations. To do so we constructed a class of well-posedness for the 1D cubic Schr\"odinger equation included in the critical Fourier-Lebesgue space $\mathcal FL^\infty$ and in supercritical Sobolev spaces with respect to scaling. Last but not least we recall that the binormal flow is a classical model for the dynamics of a vortex filament in a 3D fluid or superfluid, and that vortex motions are a key element of turbulence.