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

TL;DR

This paper investigates the stability of self-similar solutions to 1D cubic nonlinear Schrödinger equations with time-dependent coefficients, linking it to vortex filament dynamics and proving well-posedness for specific initial data.

Contribution

It introduces a stability analysis method for these solutions using pseudo-conformal transformation and constructs modified wave operators, also establishing well-posedness for certain initial conditions.

Findings

Stability results for self-similar solutions of 1D cubic NLS with time-dependent coefficients.

Construction of modified wave operators for the associated transformed equations.

Proof of well-posedness for initial data of the form z_0 / x, contrasting with Dirac delta initial data.

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.