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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-timecompact two-soliton solution with arbitrarily small $L^2$-norm which exhibitsthe following two regimes: (i) a transient turbulent regime characterized by adramatic 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 stationarylarge forever in time.

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