Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics

TLDR

A Bose‑Einstein condensate is a quantum fluid that exhibits shock‑wave–like nonlinear dynamics, yet its mean‑field Gross‑Pitaevskii equation lacks dissipation, so classical dissipative shock solutions cannot explain the observed phenomena. The study investigates whether small‑dispersion wave dynamics can generate dispersive shock waves in a Bose‑Einstein condensate. By solving the Gross‑Pitaevskii equation with small dispersion, the authors model dispersive shock wave formation and compare the results to experimental data. Numerical solutions of the Gross‑Pitaevskii equation agree with experiment, revealing distinct shock structures and front speeds versus dissipative shocks, and confirming that the observed blast waves are dispersive shock waves.

Abstract

A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock-wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of a shock wave in a compressible gas, e.g., traveling fronts with steep gradients. However, the governing Gross-Pitaevskii (GP) equation that describes the mean field of a BEC admits no dissipation, hence classical dissipative shock solutions do not explain the phenomena. Instead, wave dynamics with small dispersion is considered and it is shown that this provides a mechanism for the generation of a dispersive shock wave (DSW). Computations with the GP equation are compared to experiment with excellent agreement. A comparison between a canonical one-dimensional (1D) dissipative and dispersive shock problem shows significant differences in shock structure and shock-front speed. Numerical results associated with the three-dimensional experiment show that three- and two-dimensional approximations are in excellent agreement and 1D approximations are in good qualitative agreement. Using 1D DSW theory, it is argued that the experimentally observed blast waves may be viewed as dispersive shock waves.