Introduction to nonlinear phenomena in superfluid liquids and Bose-Einstein condensates: helium, semiconductors and graphene

We review current understanding of the non-equilibrium dynamics of collective quantum systems. We describe an approach based on the Hamiltonian formulation of superfluid hydrodynamics. It is shown that, in the presence of constant energy pumping, the nonlinear coupling of fluctuations in the density and entropy strongly affects the non-equilibrium dynamics of the system. We use the results obtained to analyse the properties of out-of-equilibrium superfluid 4 He and of exciton-polariton Bose-Einstein condensates, both in semiconductor quantum wells and in graphene layers in the presence of high magnetic field.