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
Résumé
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.
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