2023/04/21 by Valeria Banica, Banica, Valeria, Renato Lucà +5
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2304.11050
openalex publication_date 2023/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the 1D cubic NLS on \mathbb R and prove a blow-up result for functions that are of borderline regularity, i.e. Hs for any s<-\frac 12 for the Sobolev scale and \mathcal F L^∞ for the Fourier-Lebesgue scale. This is done by identifying at this regularity a certain functional framework from which solutions exit in finite time. This functional framework allows, after using a pseudo-conformal transformation, to reduce the problem to a large-time study of a periodic Schrödinger equation with non-autonomous cubic nonlinearity. The blow-up result corresponds to an asymptotic completeness result for the new equation. We prove it using Bourgain's method and exploiting the oscillatory nature of the coefficients involved in the time-evolution of the Fourier modes. Finally, as an application we exhibit singular solutions of the binormal flow. More precisely, we give conditions on the curvature and the torsion of an initial smooth curve such that the constructed solutions generate several singularities in finite time.