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

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 $L^2$-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.