We review recent work on the Calogero–Moser derivative NLS posed on the real line. This equation is found to be a completely integrable Hamiltonian PDE, whose Lax pair structure acts on the complex Hardy space \(L^2_+(\mathbb {R})\) . We give a synopsis of the current state of affairs concerning global well-posedness, its Lax pair structure, and multi-soliton solutions. In particular, we discuss the striking phenomenon of turbulence (energy cascades) for multi-solitons of this completely integrable PDE with \(L^2\) -critical scaling behaviour. Most of the results presented here are based on recent joint work of P. Gérard and the author.