Scattering for 1D cubic NLS and singular vortex dynamics

TL;DR

This paper investigates the stability and singularity formation of self-similar vortex filament solutions in fluids, using the Hasimoto transform to analyze a related 1D cubic NLS equation with time-dependent coefficients.

Contribution

It constructs solutions to the binormal flow with initial data near singular vortex filaments and proves their finite-time singularity formation, extending known local existence results.

Findings

Solutions become singular in finite time

Asymptotic completeness for the 1D cubic NLS with time-dependent coefficients

Construction of solutions with initial data close to singular vortex filaments

Abstract

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