2011/03/28 by Susana Gutierrez, Susana Gutiérrez, Luis Vega +2
Mathematics · Physics and Astronomy · #35B35 #35J10 #35Q35 #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum, superfluid, helium dynamics #math.AP #msc:35B35 #msc:35J10 #msc:35Q35 #msc:35Q55
paper · pdf · doi:10.48550/arxiv.1103.5403
38 pages, 8 figures
arxiv created 2011/03/28 · openalex publication_date 2011/03/28 · arxiv updated 2011/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 iut+uxx+(u)/(2) (|u|2-(A)/(t))=0, A∈ \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 ivt+ vxx +(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)= z0 \pv (1)/(x) for some values of z0∈ \C∖\0\, and A is adequately chosen. This is in deep contrast with the case when the initial datum is the Dirac-delta distribution.