Dark soliton solutions of (2+1)-dimensional Gross-Pitaevskii equation with spatially-periodic external potential

In this thesis, we study the existence and stability of background solutions (BGSs) and dark soliton solutions (DSSs) of the Gross-Pitaevskii equation (GPE) with a spatially-periodic external potential in (2+1)-dimensional domain. First, we use dynamical systems theory to prove the existence of stationary solutions (BGS and DSS). Exponentially decaying coefficient of the dark soliton near the BGS is derived in terms of an eigenvalue problem. The stability/instability of the solutions is investigated in a weighted functional space. The critical wave numbers in the y direction are derived. We then consider two special cases: the case of large chemical potential and the case of slowly varying external potential. In both cases, we apply asymptotic analysis to obtain the approximate formulas for the stationary solutions. As such analytic results are obtained in both cases. All the results are supported by direct numerical computations of the stationary solutions as well as the computations of the corresponding eigenvalue problems