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

Abstract

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.