On the stability of self-similar solutions of 1D cubic Schrödinger equations

In this paper we will study the stability properties of self-similar solutions of 1D cubic NLS equations with time-dependent coefficients of the form i u + u + u/2 ({pipe}u{pipe} - A/t) = 0, A ∈ ℝ. The study of the stability of these self-similar solutions is related, through the Hasimoto transformation, to the stability of some singular vortex dynamics in the setting of the Localized Induction Equation (LIE), an equation modeling the self-induced motion of vortex filaments in ideal fluids and superfluids. We follow the approach used by Banica and Vega that is based on the so-called pseudo-conformal transformation, which reduces the problem to the construction of modified wave operators for solutions of the equation iv + v + v/2t({pipe}v{pipe}-A) = 0. As a by-product of our results we prove that Eq. (0. 1) is well-posed in appropriate function spaces when the initial datum is given by u(0, x) = z p. v 1/x for some values of z ∈ ℂ\ {0}, and A is adequately chosen. This is in deep contrast with the case when the initial datum is the Dirac-delta distribution. © 2012 Springer-Verlag.