A TWO-SOLITON WITH TRANSIENT TURBULENT REGIME FOR THE CUBIC HALF-WAVE EQUATION ON THE REAL LINE

Une solution à deux solitons avec régime turbulent transitoire pour l'équation de demi-onde cubique sur la droite réelle

Patrick Gérard (1) , Enno Lenzmann (2) , Oana Pocovnicu (3) , Pierre Raphaël (4)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 Mathematisches Institut
3 HWU - Heriot-Watt University [Edinburgh]
4 LJAD - Laboratoire Jean Alexandre Dieudonné

We consider the focusing cubic half-wave equation on the real line i∂tu + |D|u = |u| 2 u, |D|u(ξ) = |ξ|û(ξ), (t, x) ∈ R+ × R. We construct an asymptotic global-in-time compact two-soliton solution with arbitrarily small L 2-norm which exhibits the following two regimes: (i) a transient turbulent regime characterized by a dramatic and explicit growth of its H 1-norm on a finite time interval, followed by (ii) a saturation regime in which the H 1-norm remains stationary large forever in time.

Mots clés: Multi-soliton; modulation theory; wave turbulence; growth of Sobolev norms; half-wave equation; cubic Szegö equation

Domaines: Equations aux dérivées partielles [math.AP]

Patrick Gérard, Enno Lenzmann, Oana Pocovnicu, Pierre Raphaël. A TWO-SOLITON WITH TRANSIENT TURBULENT REGIME FOR THE CUBIC HALF-WAVE EQUATION ON THE REAL LINE: Solution à deux solitons pour la demi-onde. 2016. ⟨hal-01403241⟩