In this talk we consider a nonlocal focusing cubic half-wave equation on the real line. Evolution problems with nonlocal dispersion naturally arise in physical settings which include models for wave turbulence, continuum limits of lattice systems, and gravitational collapse. The goal of the talk is to present the construction of an asymptotic global-in-time modulated two-soliton solution of small mass, which exhibits the following two regimes: (i) a turbulent regime characterized by an explicit growth of high Sobolev norms on a finite time interval, followed by (ii) a stabilized regime in which the high Sobolev norms remain stationary large forever in time. This talk is based on joint work with P.\ G\'erard (Orsay, France), E.\ Lenzmann (Basel, Switzerland), and P.\ Raphael (Nice, France)