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

Extended stream functions for dynamics of Bose-Einstein condensations: Snake instability of dark soliton in ultra-cold atoms as an example