In this talk I will explain what a superfluid is and introduce a simple model able to reproduce many aspects of its dynamics: the Gross—Pitaevskii (GP) equation, that is the three-dimensional nonlinear Schrödinger equation with the defocusing sign.
I will discuss why the GP equation models a superfluid and how topological defects in the form of quantised vortex lines arise naturally in this system. I will show how these coherent structures interact nonlinearly, reconnect and evolve. In particular, I will introduce the concept of superfluid turbulence (sometimes known as quantum turbulence) in this model and its similarities and differences with classical hydrodynamical turbulence. More precisely, I will explain how and why different physical phenomena emerge at different scales, focusing on the principal mechanisms responsible for transferring energy and the main results and key open questions in this area.