Blowup and scattering problems for the nonlinear Schrodinger equations

We consider L-2-supercritical and H-1-subcritical focusing nonlinear Schrodinger equations. We introduce a subset PW of H-1(R-d) for d >= 1, and investigate behavior of the solutions with initial data in this set. To this end, we divide PW into two disjoint components PW+ and PW_. Then, it turns out that any solution starting from a datum in PW+ behaves asymptotically free, and solution starting from a datum in PW_ blows up or grows up, from which we find that the ground state has two unstable directions. Our result is an extension of the one by Duyckaerts, Holmer, and Roudenko to the general powers and dimensions, and our argument mostly follows the idea of Kenig and Merle.