Mathematical Results for Nonlinear Equations of Schrödinger Type

The content of this thesis encapsulates four results in the analysis of nonlinear partial differential equations of Schrödinger type. These mathematical models are motivated by phenomena in laser optics and quantum physics, and the works themselves demonstrate several mathematical techniques frequently employed to understand properties of such models. In the first two works, we prove global well-posedness of models for laser propagation at high intensities in which off-axis variations in the group velocity and self-steepening of the pulse envelope become relevant mechanisms in preventing finite-time blow-up. The latter mechanism is considered only in the two-dimensional setting where we also provide careful numerical simulations to understand the (in)stability of such models. Of the results in quantum physics, we first consider a nonlinear Schrödinger equation (NLS) in two spatial dimensions subject to a periodic honeycomb lattice potential. By way of a multi-scale expansion together with rigorous error estimates, we derive an effective model of nonlinear Dirac type - which describes the propagation of slowly modulated, weakly nonlinear waves spectrally localized around a Dirac point. Lastly, we consider the NLS with an attractive harmonic potential and angular momentum term, which serves as a model for the mean-field dynamics of Bose-Einstein condensates in rotating harmonic traps. Here we establish several stability and instability properties for the corresponding solution, emphasizing the difference between the situation in which the trap is symmetric with respect to the rotation axis and the one where this is not the case. These works are joint collaborations in parts together with Paolo Antonelli, Christian Klein, Irina Nenciu, and my advisor Christof Sparber.