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

## TL;DR

This paper introduces extended stream functions (ESFs) to model the complex dynamics of Bose-Einstein condensates, enabling analysis of phenomena like dark soliton instabilities in ultra-cold atomic gases.

## Contribution

The paper develops ESFs that extend traditional stream functions to describe compressible, non-stationary superfluid dynamics, demonstrated through dark soliton instability analysis.

## Key findings

- ESFs successfully model superfluid dynamics beyond stationary states
- Application to dark soliton shows detailed instability behavior
- Provides a new framework for studying BEC dynamics

## Abstract

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.