Simple waves in a two-component Bose-Einstein condensate

We study the dynamics of so-called simple waves in a two-component Bose-Einstein condensate. The evolution of the condensate is described by Gross-Pitaevskii equations which can be reduced for these simple wave solutions to a system of ordinary differential equations which coincide with those derived by Ovsyannikov for the two-layer fluid dynamics. We solve the Ovsyannikov system for two typical situations of large and small difference between interspecies and intraspecies nonlinear interaction constants. Our analytic results are confirmed by numerical simulations.

Figure 1

In this plot we compare the numerical simulations for the density field $\rho (c=x/t)$ of the GP equations (1) (red lines) with the numerical solution of the Ovsyannikov equations (10) (blue lines) and the approximation (17) (black dashed lines). The vertical dotted lines indicate velocities of the edges. The initial profile is characterized by ${\rho }_1^L=1,{\rho }_2^L=0,{\rho }_1^R=0.7,{\rho }_2^R=0.3$ and the interaction constants are equal to $g_{11}=g_{22}=g=1,g_{12}=\tilde{g}=0.9$. The numerical solution of the Ovsyannikov equations confirms the approximate constancy of the total density ${\rho }_0$ (blue dashed line) in the polarization mode.

Figure 2

We compare the density field $\rho (c=x/t)$ for the numerical solution of the GP equations (1) (red lines) with the numerical solution of the Ovsyannikov equations (10) (blue lines) and the approximate analytical solution (17), (19) (black dashed lines). The vertical dotted lines indicate the edge velocities. The initial profile is characterized by ${\rho }_1^L=0.742,{\rho }_2^L=0,{\rho }_1^R=0.7,{\rho }_2^R=0.3$ and the interaction constants are equal to $g_{11}=g_{22}=g=1,g_{12}=\tilde{g}=0.1$. The total density ${\rho }_0$ obtained by the numerical solution of the Ovsyannikov equations is shown by a blue dashed line. It is clear that this density is inhomogeneous in this case.

Figure 3

In this plot we compare the numerical simulations for the density field $\rho (c=x/t)$ obtained by numerical solution of the GP equations (1) (red lines) with the approximate solution (17), (19), (30), (33), (34) (blue lines). The vertical dotted lines denote the velocities of the edges. The initial profile is characterized by ${\rho }_1^L=1,{\rho }_2^L=0,{\rho }_1^R=0.7,{\rho }_2^R=0.3$ and the interaction constants are equal to $g_{11}=g_{22}=g=1,g_{12}=\tilde{g}=0.1$. The wave structure in the first component consists of a dispersive shock wave on the right, two connected rarefaction waves on the left, and a plateau between them. The second component has a single rarefaction wave only.