Cascade of phase shifts for nonlinear Schrodinger equations

2005/02/11 by Rémi Carles, Remi Carles, Carles, Remi
Engineering · Mathematics · #35C20 #35Q55 #81Q20 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #math.AP #msc:35C20 #msc:35Q55 #msc:81Q20

paper · pdf · doi:10.48550/arxiv.math/0502242

a4 paper, 18 pages, 1 figure

arxiv created 2005/02/11 · openalex publication_date 2005/02/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a semi-classical nonlinear Schrodinger equation. For initial data causing focusing at one point in the linear case, we study a nonlinearity which is super-critical in terms of asymptotic effects near the caustic. We prove the existence of infinitely many phase shifts appearing at the approach of the critical time. This phenomenon is suggested by a formal computation. The rigorous proof shows a quantitatively different asymptotic behavior. We explain these aspects, and discuss some problems left open.

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