Geodesic
Scattering for 1D cubic NLS and singular vortex dynamics
Journal of the European Mathematical Society, Tome 14 (2012) no. 1, pp. 209-253

Voir la notice de l'article provenant de la source EMS Press

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 χa​(t,x) form a family of evolving regular curves in R3 that develop a singularity in finite time, indexed by a parameter a>0. We consider curves that are small regular perturbations of χa​(t0​,x) for a fixed time t0​. 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.
DOI : 10.4171/jems/300
Classification : 76-XX, 35-XX, 00-XX
Keywords: Vortex filaments, binormal flow, selfsimilar solutions, Schr¨odinger equations, scattering 
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     title = {Scattering for {1D} cubic {NLS} and singular vortex dynamics},
     journal = {Journal of the European Mathematical Society},
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Valeria Banica; Luis Vega. Scattering for 1D cubic NLS and singular vortex dynamics. Journal of the European Mathematical Society, Tome 14 (2012) no. 1, pp. 209-253. doi: 10.4171/jems/300

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