Dissipationless shock waves in Bose-Einstein condensates with repulsive interaction between atoms

We consider formation of dissipationless shock waves in Bose-Einstein condensates with repulsive interaction between atoms. It is shown that for big enough initial inhomogeneity of density, interplay of nonlinear and dispersion effects leads to wave breaking phenomenon followed by generation of a train of dark solitons. Analytical theory is confirmed by numerical simulations.

Figure 1

Wave breaking of a boxlike initial density distribution (shown by dashed line) in the hydrodynamic approximation. The plot corresponds to the solution (6) with $a=10,$ ${\rho }_0=1,$ ${\rho }_1=10,$ and $t=1$. Point $A$ propagates inward the box with local sound velocity $2\sqrt{{\rho }_1}$, point $C$ propagates outward along the background density with local sound velocity $2\sqrt{{\rho }_0}$, and point $B$ corresponds to the intersection of two simple wave solutions with profiles $AB$ and $CB$.

Figure 2

Formation of dissipationless shock wave after wave breaking point according to Whitham modulation theory applied to the 1D NLS equation. The dashed line corresponds to the multivalued region arising in the hydrodynamic approximation given by Eq. (7) (it is analogous to the region between the points B and C in Fig. 1), and the solid line represents the modulated periodic wave given by Eqs. (8, 11). Both profiles are calculated for $\overline{\lambda }=-10$ at time $t=1$.

Figure 3

(a) Two-dimensional initial distribution of BEC density with paraboloid hump on a constant background given by Eq. (19) with $a=10,$ ${\rho }_0=1,$ and ${\rho }_1=10$. (b) Two-dimensional density distribution of BEC after time $t=2$ of the evolution from the initial paraboloid density on constant background according to a numerical solution of the 2D NLS equation (18).

Figure 4

Cross sections of the density profile at different evolution time according to the numerical solution of the 2D NLS equation (18) with the initial condition (19).