Hydrodynamics of cold atomic gases in the limit of weak nonlinearity, dispersion, and dissipation

Dynamics of interacting cold atomic gases have recently become a focus of both experimental and theoretical studies. Often cold-atom systems show hydrodynamic behavior and support the propagation of nonlinear dispersive waves. Although this propagation depends on many details of the system, great insight can be obtained in the rather universal limit of weak nonlinearity, dispersion, and dissipation. In this limit, using a reductive perturbation method we map some of the hydrodynamic models relevant to cold atoms to well-known chiral one-dimensional equations such as the Korteweg-de Vries (KdV), Burgers, KdV-Burgers, and Benjamin-Ono equations. These equations have been thoroughly studied in the literature. The mapping gives us a simple way to make estimates for the original hydrodynamic equations and to study the interplay between nonlinearity, dissipation, and dispersion which are the hallmarks of nonlinear hydrodynamics.