On a one-dimensional Gross-Pitaevskii-type model for a condensate past an obstacle: dynamics and stability properties of solutions

A remarkable phenomenon that Bose-Einstein condensates (BECs) exhibit is superfluidity. A classical way to experimentally probe this phenomenon consists in studying the response of a BEC to a moving impurity. In this case, superfluidity manifests in the existence of a critical velocity below which the impurity can move without exchanging energy or momentum with the fluid. In the present thesis we use a simple but effective mathematical model to investigate this phenomenon. It is based on the one-dimensional Gross-Pitaevskii equation with non-zero boundary conditions at infinity. An external potential, that travels at constant speed, is added to the equation to account for the presence of the moving impurity. Within this model, superfluidity manifests in the existence of a critical velocity below which the system admits solutions which are stationary in the impurity's reference frame. In the first part of the thesis, we model the impurity by a repulsive delta potential. In a first instance, we study the global well-posedness of the model in the natural energy space of the system. A non-trivial difficulty in this step lies in the definition of the conserved energy. Due to the non-zero boundary conditions at infinity, its definition involves the notion of renormalized momentum, which we carefully discuss. Next, we study, in a co-moving reference frame, the existence and stability of stationary solutions. For any positive strength of the delta potential, the following bifurcation diagram was already known: for velocities below a critical value, there are two branches of stationary solutions; these merge at the critical velocity, and disappear in the supercritical regime. The stability of the two branches has been investigated by numerical simulations, suggesting that the branch which is more energetic is unstable, while the branch which is less energetic is stable. By means of a variational argument, we show the orbital stability of the least energetic branch. In the second part of the thesis we study the case in which the impurity is modeled by a weak external potential. We study the existence and stability of stationary solutions. For any given non-zero velocity in the subsonic regime, and under a non-degeneracy condition, we show the existence of a linearly unstable branch of stationary solutions, this time indexed by the strength of the external potential $\varepsilon$. The existence result follows from a Lyapunov-Schmidt reduction method. The instability analysis, instead, is performed by employing an Evans function approach. Finally, we formally apply the above instability result to the case of a weak delta potential. We address in this way the stability properties of the most energetic branch of stationary solutions, and we obtain a linear instability result.