# Scattering for 1D cubic NLS and singular vortex dynamics

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 1, page 209-253
- ISSN: 1435-9855

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topBanica, 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 -

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