Large data scattering for the defocusing supercritical generalized KdV equation

We consider the defocusing supercritical generalized Korteweg-de Vries (gKdV) equation ${\mathrm{\partial }}_{t}u+{\mathrm{\partial }}_x^{3}u-{\mathrm{\partial }}_x(u^{k+1})=0$, where $k>4$ is an even integer number. We show that if the initial data $u_0$ belongs to $H^1$ then the corresponding solution is global and scatters in $H^1$. Our method of proof is inspired on the compactness method introduced by C. Kenig and F. Merle.