Solitons in Exciton-Polariton System: Reduced Modelling, Analysis, and Numerical Studies

Abstract

In this dissertation, we study the behavior of exciton-polariton quasiparticles in semiconductor microcavities, under the sourceless and lossless conditions.First, we simplify the original model by removing the photon dispersion term, thus effectively turn the PDEs system to an ODEs system, and investigate the behavior of the resulting system, including the equilibrium points and the wave functions of the excitons and the photons. Second, we add the dispersion term for the excitons to the original model and prove that the band of the discontinuous solitons now become dark solitons. Third, we employ the Strang-splitting method to our sytem of PDEs and prove the first-order and second-order error bounds in the $H^1$ norm and the $L_2$ norm, respectively. Using this numerical result, we analyze the stability of the steady state bright soliton solution. This solution revolves around the $x$-axis as time progresses and the perturbed soliton also rotates around the $x$-axis and tracks closely in terms of amplitude but lags behind the exact one. Our numerical result shows orbital stability but no $L_2$ stability.