Affiliation: College of Arts and Sciences, Department of Mathematics
In studying the cubic nonlinear Schrödinger (NLS) equation with hexagonal lattice potential, Ablowitz, Nixon, and Zhu  and Fefferman and Weinstein  have used ansatz solutions of a periodic, cubic NLS equation to derive two similar two-dimensional Dirac equations with cubic nonlinearity. Chapters 1 and 2 of this thesis deal with solutions and lifespans of solutions for the linear and nonlinear Dirac equations. We establish local and almost global existence results as well as ill-posedness below the critical regularity, H ̇ 1/2. We leave as an open question whether solutions blow up in finite time or if a global existence result can be found. The third chapter modifies the machinery of  and  to explore an open question posed by Fefferman and Weinstein, . We prove that an envelope of solutions to the slowly modulated Dirac equation provides a good approximation for solutions to the nonlinear Schrödinger equation with hexagonal lattice. The NLS solution is shown to exist for long times with the nonlinear Dirac dynamics affecting the solution on any constant timescale. The same timescale is also proven for an ansatz envelope proposed by Ablowitz, Nixon, and Zhu. The timescale is also extended in an intermediate regime by slightly weakening the nonlinearity of the governing Dirac equation. The final chapter discusses future work focusing on some numerical simulations for the Dirac and Schrödinger equations.