Nonlinear aspects of the theory of Bose-Einstein condensates

The study of imperfect Bose gases provides an opportunity to understand how irreversibility arises in superflows, as it is a unique dynamical model where hydrodynamics can be related in a rational way to microscopic quantum dynamics. Over the years various specific problems have been investigated in this field, all with interesting implications in nonlinear science. Within the framework of kinetic theory we first review the way the condensate appears - an interesting example of finite-time blow-up of solutions of a nonlinear equation, with exponents derived from a nonlinear eigenvalue problem. Then we examine how the presence of the condensate changes the kinetic theory - something that is not yet fully understood. Finally, we look at the phenomenon of vortex nucleation and release in a model of superflow. This interesting example of a transonic transition can be analysed by solving locally the equation of compressible (super)fluid mechanics near the elliptic/hyperbolic transition.