2009/03/31 by Fabrice Béthuel, Fabrice Bethuel, Philippe Gravejat +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Cold Atom Physics and Bose-Einstein Condensates #Gross–Pitaevskii equation #Korteweg–de Vries equation #Limit (mathematics) #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear system #Physics #Quantum mechanics #Traveling wave #math.AP #msc:35Q53 #msc:35Q55
paper · pdf · doi:10.1080/03605300903222542
published as Communications in Partial Differential Equations 35, 1 (2010) 113-164 · Final version accepted for publication in Communications in Partial Differential Equations with a few minor corrections and added remarks
arxiv created 2009/12/12 · openalex publication_date 2009/12/12 · arxiv updated 2010/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper, we proceed along our analysis of the Korteweg–de Vries approximation of the Gross–Pitaevskii equation initiated in [Citation6]. At the long-wave limit, we establish that solutions of small amplitude to the one-dimensional Gross–Pitaevskii equation split into two waves with opposite constant speeds , each of which are solutions to a Korteweg–de Vries equation. We also compute an estimate of the error term which is somewhat optimal as long as travelling waves are considered. At the cost of higher regularity of the initial data, this improves our previous estimate in [Citation6].