Global existence for the critical generalized KdV equation

We discuss results regarding global existence of solutions for the critical generalized Korteweg-de Vries equation, \[ u t + u x x x + u 4 u x = 0 , x , t ∈ R . u_t+u_{xxx}+u^4\,u_x=0,\quad x,\,t\in \mathbb {R}. \] The theory established shows the existence of global solutions in Sobolev spaces with order below the one given by the energy space H 1 ( R ) H^1(\mathbb {R}) , i.e. solutions corresponding to data u 0 ∈ H s ( R ) u_0\in H^s(\mathbb {R}) , s > 3 / 4 s>3/4 , with ‖ u 0 ‖ L 2 > ‖ Q ‖ L 2 \|u_0\|_{L^2}>\|Q\|_{L^2} , where Q Q is the solitary wave solution of the equation.