Scattering for 1D cubic NLS and singular vortex dynamics

Banica, Valeria, and Vega, Luis. "Scattering for 1D cubic NLS and singular vortex dynamics." Journal of the European Mathematical Society 014.1 (2012): 209-253. <http://eudml.org/doc/277492>.

@article{Banica2012,
abstract = {We study the stability of 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 in $\mathbb \{R\}^3$ 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 solutions of the binormal flow with these initial data. Moreover, these solutions become also singular in finite time. Our approach uses the Hasimoto transform, which 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.},
author = {Banica, Valeria, Vega, Luis},
journal = {Journal of the European Mathematical Society},
keywords = {vortex filaments; binormal flow; selfsimilar solutions; Schrödinger equations; scattering; vortex filaments; binormal flow; selfsimilar solutions; Schrödinger equations; scattering},
language = {eng},
number = {1},
pages = {209-253},
publisher = {European Mathematical Society Publishing House},
title = {Scattering for 1D cubic NLS and singular vortex dynamics},
url = {http://eudml.org/doc/277492},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Banica, Valeria
AU - Vega, Luis
TI - Scattering for 1D cubic NLS and singular vortex dynamics
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 1
SP - 209
EP - 253
AB - We study the stability of 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 in $\mathbb {R}^3$ 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 solutions of the binormal flow with these initial data. Moreover, these solutions become also singular in finite time. Our approach uses the Hasimoto transform, which 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.
LA - eng
KW - vortex filaments; binormal flow; selfsimilar solutions; Schrödinger equations; scattering; vortex filaments; binormal flow; selfsimilar solutions; Schrödinger equations; scattering
UR - http://eudml.org/doc/277492
ER -