On the stability of self-similar solutions of 1D cubic Schrödinger 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 0.1 $$\begin{aligned} \displaystyle { iu_t+u_{xx}+\frac{u}{2} \left(|u|^2-\frac{A}{t}\right)=0, \quad A\in \mathbb{R }. } \end{aligned}$$ 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 $$\begin{aligned} iv_t+ v_{xx} +\frac{v}{2t}(|v|^2-A)=0. \end{aligned}$$ 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 $$u(0,x)= z_0 \mathrm p.v \frac{1}{x}$$ for some values of $$z_0\in \mathbb{C }\setminus \{ 0\}$$ , and $$A$$ is adequately chosen. This is in deep contrast with the case when the initial datum is the Dirac-delta distribution.