  In this paper, we formulate extended stream functions (ESFs) to describe the
dynamics of Bose-Einstein condensations in the two-dimensional space. The
ordinary stream function is applicable only for stationary and incompressible
superfluids, whereas the ESFs can describe the dynamics of compressible and
non-stationary superfluids. The ESFs are composed of two stream functions,
i.e., one describes the compressible density modulations and the other the
incompressible rotational superflow. As an application, we study the snake
instability of the dark soliton in a rectangular potential in detail by the
ESFs.
