Asymptotic Stability for Some Non positive Perturbations of the Camassa–Holm Peakon with Application to the Antipeakon–peakon Profile. (2nd October 2018)

• Record Type: Journal Article
• Authors: Molinet, Luc
• Is Part Of: International mathematics research notices. Volume 2020:Number 21(2020)
• Page Start: 7908
• Page End: 7943
• Publication Date: 2018-10-02
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rny224
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: We continue our investigation on the asymptotic stability of the peakon. In a 1st step we extend our asymptotic stability result [30 ] in the class of functions whose negative part of the momentum density is supported in $]-\infty, x_0] $ and the positive part in $[x_0, +\infty [$ for some $ x_0\in{\mathbb{R}}$ . In a 2nd step this enables us to prove the asymptotic stability of well-ordered train of antipeakons–peakons and, in particular, of the antipeakon–peakon profile. Finally, in the appendix we prove that in the case of a nonnegative momentum density the energy at the left of any given point decays to zero as time goes to $+\infty $ . This leads to an improvement of the asymptotic stability result stated in [30 ].