| Nonlinear dispersive equations model nonlinear waves in a wide range of physical and mathematics contexts. They reinforce or dissipate effects of linear dispersion and nonlinear interactions, and thus, may be of a focusing or defocusing nature. The nonlinear Schrodinger equation or NLS is an example of such equations. It appears as a model in hydrodynamics, nonlinear optics, quantum condensates, heat pulses in solids and various other nonlinear instability phenomena. In mathematics, one of the interests is to look at the wave interaction: waves propagation with different speeds and/or different directions produces either small perturbations comparable with linear behavior, or creates solitary waves, or even leads to singular solutions.;This dissertation studies the global behavior of finite energy solutions to the d-dimensional focusing NLS equation, i∂ tu + Deltau + |u| p-1u = 0, with initial data u0 ∈ H1, x ∈ Rd ; the nonlinearity power p and the dimension d are chosen so that the scaling index s = d2-2p-1 is between 0 and 1, thus, the NLS is mass-supercritical ( s > 0) and energy-subcritical (s < 1).;For solutions with ME [u0] < 1 ( ME [u0] stands for an invariant and conserved quantity in terms of the mass and energy of u0), a sharp threshold for scattering and blowup is given. Namely, if the renormalized gradient Gu of a solution u to NLS is initially less than 1, i.e., Gu (0) < 1, then the solution exists globally in time and scatters in H1 (approaches some linear Schrodinger evolution as t → +/-infinity); if the renormalized gradient Gu (0) > 1, then the solution exhibits a blowup behavior, that is, either a finite time blowup occurs, or there is a divergence of H1 norm in infinite time.;This work generalizes the results for the 3d cubic NLS obtained in a series of papers by Holmer-Roudenko and Duyckaerts-Holmer-Roudenko with the key ingredients, the concentration compactness and localized variance, developed in the context of the energy-critical NLS and Nonlinear Wave equations by Kenig and Merle.;One of the difficulties is fractional powers of nonlinearities which are overcome by considering Besov-Strichartz estimates and various fractional differentiation rules. |
