Global rough solutions to the cubic nonlinear Boussinesq equation

We prove that the initial value problem (IVP) for the cubic defocusing nonlinear Boussinesq equation $u_{tt}-u_{xx}+u_{xxxx}-(\mathrm{\mid }u{\mathrm{\mid }}^{2}u{)}_{xx}=0$ on the real line is globally well-posed in $H^s(\mathbb{R})$ provided $2\mathrm{/}3<s<1$.