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A Two-Soliton with Transient Turbulent Regime for the Cubic Half-Wave Equation on the Real Line

Accepted version
Peer-reviewed

Type

Article

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Authors

Gérard, Patrick 
Lenzmann, Enno 
Pocovnicu, Oana 
Raphaël, Pierre 

Abstract

We consider the focusing cubic half-wave equation on the real line $$ i \partial_t u + |D| u = |u|^2 u, \ \ \widehat{|D|u}(\xi)=|\xi|\hat{u}(\xi), \ \ (t,x)\in \Bbb R_+\times \Bbb R. $$ We construct an asymptotic global-in-time compact two-soliton solution with arbitrarily small L2-norm which exhibits the following two regimes: (i) a transient turbulent regime characterized by a dramatic and explicit growth of its H1-norm on a finite time interval, followed by (ii) a saturation regime in which the H1-norm remains stationary large forever in time.

Description

Keywords

4901 Applied Mathematics, 49 Mathematical Sciences

Journal Title

Annals of PDE

Conference Name

Journal ISSN

2524-5317
2199-2576

Volume Title

4

Publisher

Springer

Rights

All rights reserved
Sponsorship
European Commission Horizon 2020 (H2020) ERC (SINGWAVES 646650)
ERC consolidator grant