Topological solitons of the nonlinear Dirac equation in 2+1 dimensional space time - Dikole, Realeboga (Department of Mathematics and Applied Mathematics, 2024)
The work of this thesis is a presentation of nonlinear Dirac-type models with the primary focus being on planar, (2 + 1) dimensional nonlinear Dirac models.We study a (2 + 1) dimensional extension of the (2 + 0) dimensional reduction the complex sine-Gordon. This is a tachyonic nonlinear Dirac equation whose linear part can be reduced to the imaginary mass Klein-Gordon equation. Although this model is tachyonic it can be restored into a real and non-hypothetical version by considering it in nonvanishing backgrounds. We investigate the stability of the single vortex solution by considering perturbation about the single vortex solution. Perturbations include the radially symmetric perturbations (m = 0) and angular perturbations (m ∈ {±1,±2}). The single vortex was found to be stable for both the radially symmetric and angular perturbations m = {0,±1,±2}, with the real part of the eigenvalues having a negligible nonzero real part of order 10−3. The eigenvalues presented were obtained by use of the sine series expansion and Chebyshev spectral method, where the author is the first to present this work. The Chebyshev spectral method was found to outperform the sine series expansion in terms of computation times. However, the drawback of Chebyshev differentiation matrices is that they contain spurious eigenvalues that grow proportional to the number of modes N. We also present the planar Soler model, both tachyonic and tardyonic and find that the planar tachyonic Soler model does not admit stationary vortex solutions. On the other hand, the tardyonic Soler model possesses stable vortex solutions as shown by Cuevas-Maraver et al. We also study the numerical solution of another planar nonlinear Dirac model i.e. the nonlinear Dirac equation with Kerr nonlinearity. The (2 + 1) dimensional nonlinear Dirac equation with Kerr nonlinearity admits stationary vortex solutions as was shown in the work of Smirnova et al. Moreover, the (2+1) dimensional nonlinear Dirac equation with Kerr nonlinearity supports topological edge states.