JournalsjemsVol. 14, No. 1pp. 209–253

Scattering for 1D cubic NLS and singular vortex dynamics

  • Luis Vega

    Universidad del Pais Vasco, Bilbao, Spain
  • Valeria Banica

    Université d'Evry - Val d'Essonne, France
Scattering for 1D cubic NLS and singular vortex dynamics cover
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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 R3\mathbb R^3 that develop a singularity in finite time, indexed by a parameter a>0a > 0. We consider curves that are small regular perturbations of χa(t0,x)\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.

Cite this article

Luis Vega, Valeria Banica, Scattering for 1D cubic NLS and singular vortex dynamics. J. Eur. Math. Soc. 14 (2012), no. 1, pp. 209–253

DOI 10.4171/JEMS/300