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

• Published in: Mathematische annalen Vol. 356; no. 1; pp. 259 - 300
• Main Authors: Gutierrez, S., Vega, L.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.05.2013
• Subjects: Mathematics; Mathematics and Statistics; 35J10; 35Q55; 35B35; 35Q35
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-012-0847-4

In this paper we will study the stability properties of self-similar solutions of D cubic NLS equations with time-dependent coefficients of the form 0.1 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 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 for some values of , and is adequately chosen. This is in deep contrast with the case when the initial datum is the Dirac-delta distribution.