Scattering for 1D cubic NLS and singular vortex dynamics

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 χa(t, x) form a family of evolving regular curves of R 3 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 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.

On the Relationship Between the One-Corner Problem and the M-Corner Problem for the Vortex Filament Equation

“…which is the simplest case, has been proven in [20] (see also [12] for the corresponding hyperbolic space problem); and numerical simulations of these solutions have been carried out in [8,13]. Furthermore, the fact that this kind of solutions yields a well-posed problem has been shown in a series of papers [2,3,4,5]; in particular, [5], closes the question, because it proves that the problem with one-corner initial data is well-posed in an adequate function space. Along this paper, all the vectors are given in column form.…”

Vortex filament equation for a regular polygon

“…Recovering X and T from ψ at t = t pq Given a function ψ(s, t pq ) = α(s, t pq ) + iβ(s, t pq ), recovering the basis vectors T, e 1 and e 2 from ψ implies integrating (22). As seen in the previous section, at t = t pq , ψ(s, t) is a sum of M q (if q odd) or M q/2 (if q even) equally spaced Dirac deltas in s ∈ [0, 2π), that corresponds to a skew polygon X(s, t pq ) of M q or M q/2 sides.…”

“…Therefore, we work with data in critical spaces and the problem turns out to be much more involved. The case of an infinite curve with just one corner and that is otherwise smooth has been studied in the sequence of papers [22,23,24] by Banica and Vega. One of the conclusions is that (6) is in fact ill-posed for this type of initial conditions, so that (1) is also ill-posed if it is understood in a classical sense.…”

Motion of a vortex filament on a slanted plane

“…Koiso [7] constructed a transformation, sometimes referred to as the generalized Hasimoto transformation, and gave a mathematically rigorous proof of the equivalence of the LIE and (1.4). More recently, Banica and Vega [2,3] and Gutiérrez, Rivas, and Vega [4] constructed and analyzed a family of self-similar solutions of the LIE which forms a corner in finite time. The authors [1] proved the unique solvability of an initial-boundary value problem for the LIE in which the filament moved in the threedimensional half space.…”