  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.
