We consider the one dimensional cubic nonlinear Shr"odinger equation (NLS): $$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$ $$\phi (x,0) = \phi_{0}(x)\in H^{s}(\mathbb R);$$ where $H^{s}(\mathbb R)$ is usual Sobolev space.

In 1978 Giniberg-Velo have shown that the above NLS is globally wellposed: that is, for the initial data in $\phi_{0}\in H^{1}(\mathbb R)$, the NLS has a uniqe solution in $C(\mathbb R, H^{1}(\mathbb R)).$

My Question: Let $\phi_{0}\in H^{s}(\mathbb R), (0<s<1).$ Then what can we say about the local and global well posedness of the above NLS ? (If it is well know just the proper refeence will be o.k for me)

Thanks,