Nonlinear dispersive equations describe wave evolution with strong interactions in various physical contexts, such as plasma, superfluid, and water waves. Each equation typically produces many types of solutions, such as scattering, solitons, and blow-up, by competition between the dispersion and the interactions. Recent progress in the space-time analysis, combined with the variational arguments as well as those in the dynamical system and in the spectral theory, has enabled us to study global behavior of large solutions, but the relation between different types is still to be explored.
In this talk, we consider the nonlinear Schrodinger equation in three dimensions with attractive linear potential and nonlinear interaction, as a simple model case with the four typical solutions: scattering, blow-up, stable solitons and unstable solitons. Restricting the solutions by small mass and energy slightly above the first excited states, we can classify them into 9 sets by global behavior. The blow-up solutions are separated from the solutions scattering to the ground states, by an invariant manifold of codimension one, which is around translations of the potential-free ground state. The transverse intersection of the manifold and its time inversion gives the nine-set decomposition. The dynamic transition from the scattering to the blow-up is stable, taking place near the first excited states once and for all.