TP02: On Bose–Einstein condensates with spatially inhomogenous scattering length

I investigated a Bose–Einstein condensate confined harmonically to 1D, within the framework of Gross–Pitaevskii mean–field theory. I consider the 1D Gross–Pitaevskii equation with a combined step in its external potential and nonlinear terms. I generalise solitary wave solutions to the equation in its nonlinear limit, to take into account a constant potential and nonlinearity. I then generalise an e↵ective potential theory to take into account the combined step. The theory allows us to predict the linear stability eigenvalues of the Bogoliubov–de Gennes equations. The theoretical work is supplemented by numerical investigations of the solitary wave solutions, mainly the e↵ect of varying the step width on the solitary wave solutions. It is found that dark solitary waves are unstable for all step widths: either with positive imaginary eigenvalues; or with a complex eigenvalue quartet. Bright solitary waves undergo a pitchfork bifurcation from stability to instability (or vice–versa, depending on the sign of the step) as the step width alters. The size of the relevant eigenvalues are predicted well by the theory, becoming quantitatively less accurate as the step strength is increased, but maintaining their qualitative accuracy. Finally the time–development of an unstable solitary wave is shown. It is found that the solitary wave leaves the step region, emitting dispersive waves as it crosses the step edge. It is hoped that this work will contribute to ongoing investigations into Bose–Einstein condensates with spatially inhomogenous scattering length.