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We analyze numerically the formation and the subsequent dynamics oftwo-dimensional matter wave dark solitons in a Thomas-Fermi rubidium condensateusing various techniques. An initially imprinted sharp phase gradient leads tothe dynamical formation of a stationary soliton as well as very shallow greysolitons, whereas a smooth gradient only creates grey solitons. The depth andhence, the velocity of the soliton is provided by the spatial width of thephase gradient, and it also strongly influences the snake-instability dynamicsof the two dimensional solitons. The vortex dipoles stemming from the unstablesoliton exhibit rich dynamics. Notably, the annihilation of a vortex dipole viaa transient dark lump or a vortexonium state, the exchange of vortices betweeneither a pair of vortex dipoles or a vortex dipole and a single vortex, and soon. For sufficiently large width of the initial phase gradient, the solitonsmay decay directly into vortexoniums instead of vortex pairs, and also thedecay rate is augmented. Later, we discuss alternative techniques to generatedark solitons, which involve a Gaussian potential barrier and time-dependentinteractions, both linear and periodic. The properties of the solitons can becontrolled by tuning the amplitude or the width of the potential barrier. Inthe linear case, the number of solitons and their depths are determined by thequench time of the interactions. For the periodic modulation, a transientsoliton lattice emerges with its periodicity depending on the modulationfrequency, through a wave number selection governed by the local Bogoliubovspectrum. Interestingly, for sufficiently low barrier potential, both Faradaypattern and soliton lattice coexist. The snake instability dynamics of thesoliton lattice is characteristically modified if the Faraday pattern ispresent.

Dark solitons generation and their instability dynamics in two dimensional condensates