High-speed excited multi-solitons in nonlinear Schrödinger equations

• Published in: Journal de mathématiques pures et appliquées 2011-08, Vol.96 (2), p.135-166
• Main Authors: Côte, Raphaël, Le Coz, Stefan
• Format: Article
• Language: English
• Subjects: Analysis of PDEs; Exact sciences and technology; Excited states; General mathematics; General, history and biography; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Multi-solitons; Nonlinear Schrödinger equations; Partial differential equations; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0021-7824
• DOI: 10.1016/j.matpur.2011.03.004

We consider the nonlinear Schrödinger equation in R d i ∂ t u + Δ u + f ( u ) = 0 . For d ⩾ 2 , this equation admits traveling wave solutions of the form e i ω t Φ ( x ) (up to a Galilean transformation), where Φ is a fixed profile, solution to − Δ Φ + ω Φ = f ( Φ ) , but not the ground state. This kind of profiles are called excited states. In this paper, we construct solutions to NLS behaving like a sum of N excited states which spread up quickly as time grows (which we call multi-solitons). We also show that if the flow around one of these excited states is linearly unstable, then the multi-soliton is not unique, and is unstable. On considère lʼéquation de Schrödinger non-linéaire dans R d i ∂ t u + Δ u + f ( u ) = 0 . Pour d ⩾ 2 , cette équation admet des ondes progressives de la forme e i ω t Φ ( x ) (à une transformation galiléenne près), où Φ est un profil fixe, solution de − Δ Φ + ω Φ = f ( Φ ) , mais pas un état fondamental. Ces profils sont appelés états excités. Dans cet article, on construit des solutions de NLS se comportant comme une somme dʼétats excités qui se séparent rapidement au cours du temps (on les appelle multi-solitons). On montre aussi que si le flot autour dʼun des états excités est linéairement instable, alors le multi-soliton nʼest pas unique et est instable.