# Wave Breaking in Dispersive Fluid Dynamics of the Bose-Einstein Condensate

## TL;DR

This paper analyzes wave breaking and dispersive shock wave formation in Bose-Einstein condensates using the Gross-Pitaevskii equation, providing explicit solutions for specific initial profiles and shock velocities.

## Contribution

It derives closed-form solutions for the Whitham modulation equations for initial profiles with power-law forms, advancing understanding of wave breaking in BECs.

## Key findings

- Closed-form solutions for n=2,3 cases.
- Edge velocities of shock waves for arbitrary n>1.
- Application to experimental BEC wave phenomena.

## Abstract

The problem of wave breaking during its propagation in the Bose-Einstein condensate to a stationary medium is considered for the case when the initial profile at the breaking instant can be approximated by a power function of the form $(-x)^{1/n}$. The evolution of the wave is described by the Gross-Pitaevskii equation so that a dispersive shock wave is formed as a result of breaking; this wave can be represented using the Gurevich-Pitaevskii approach as a modulated periodic solution to the Gross-Pitaevskii equation, and the evolution of the modulation parameters is described by the Whitham equations obtained by averaging the conservation laws over fast oscillations in the wave. The solution to the Whitham modulation equations is obtained in closed form for $n = 2,3$, and the velocities of the dispersion shock wave edges for asymptotically long evolution times are determined for arbitrary integer values $n > 1$. The problem considered here can be applied for describing the generation of dispersion shock waves observed in experiments with the Bose-Einstein condensate.