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. 
Patrick G\'erard (LM-Orsay), Enno Lenzmann, Oana Pocovnicu, Pierre   Rapha\"el (JAD)
math.AP