  In this paper we will study the stability properties of self-similar
solutions of 1-d cubic NLS equations with time-dependent coefficients of the
form iu_t+u_{xx}+\frac{u}{2} (|u|^2-\frac{A}{t})=0, A\in \R (cubic). 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_t+ v_{xx}
+\frac{v}{2t}(|v|^2-A)=0. As a by-product of our results we prove that equation
(cubic) is well-posed in appropriate function spaces when the initial datum is
given by u(0,x)= z_0 \pv \frac{1}{x} for some values of z_0\in
\C\setminus\{0\}, and A is adequately chosen. This is in deep contrast with the
case when the initial datum is the Dirac-delta distribution.
