English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper we study the stability of the self-similar solutions of thebinormal flow, which is a model for the dynamics of vortex filaments in fluidsand super-fluids. These particular solutions $\chi_a(t,x)$ form a family ofevolving regular curves of $\mathbb R^3$ that develop a singularity in finitetime, indexed by a parameter $a>0$. We consider curves that are small regularperturbations of $\chi_a(t_0,x)$ for a fixed time $t_0$. In particular, theircurvature is not vanishing at infinity, so we are not in the context of knownresults of local existence for the binormal flow. Nevertheless, we construct inthis article solutions of the binormal flow with these initial data. Moreover,these solutions become also singular in finite time. Our approach uses theHasimoto transform what leads us to study the long-time behavior of a 1D cubicNLS equation with time-depending coefficients and small regular perturbationsof the constant solution as initial data. We prove asymptotic completeness forthis equation in appropriate function spaces.

Scattering for 1D cubic NLS and singular vortex dynamics