Nonlinear waves in two-component Bose-Einstein condensates: Manakov system and Kowalevski equations

Traveling waves in two-component Bose-Einstein condensates whose dynamics is described by the Manakov limit of the Gross-Pitaevskii equations are considered in a general situation with relative motion of the components when their chemical potentials are not equal to each other. It is shown that in this case the solution is reduced to the form known in the “Kowalevski top” theory of motion. Typical situations are illustrated by the particular cases when the general solution can be represented in terms of elliptic functions and their limits. Depending on the parameters of the wave, both density waves (with in-phase motions of the components) and polarization waves (with counterphase motions) are considered.

Figure 1

Regions of variations of the parameters $q_1$ and $q_2$ [see Eq. (12)] corresponding to the conditions of positivity of the densities ${\rho }_1$ and ${\rho }_2$ defined by Eqs. (8).

Figure 2

Dependence of the total density $\rho \left(\xi \right)$ (a) and the component densities ${\rho }_1\left(\xi \right)$ (b) and ${\rho }_2\left(\xi \right)$ (c) on $\xi$ in a “quasisoliton.” Parameters are equal to $V=0.5,r_1=0.857,r_2=r_3=1.0,\theta =2.5,{\theta }_2=0.5$.

Figure 3

Plots of the densities of the components ${\rho }_1$ and ${\rho }_2$ as functions of $\xi$ [see Eqs. (32)]. In all plots $u_{10}=0,u_{20}=0.5$, and $V=1.0$. In panel (a) ${\rho }_{10}=0.52,{\rho }_{20}=0.48$, and in panel (b) ${\rho }_{10}=0.48,{\rho }_{20}=0.52$.

Figure 4

Dependence of the inverse width $\kappa$ of a dark-dark soliton on its velocity $V$ for different values of the relative velocity between condensates. In all plots ${\rho }_{10}={\rho }_{20}=0.5,u_{10}=0$, and $V$ is measured with respect to quiescent condensate: (a) $u_{20}=0$, this is the case of well-known dark-dark solitons propagating through two still condensates; (b) plot corresponding to $u_{20}=1.5$ illustrates a nonmonotonic dependence; (c) at $u_{20}=2.0$ the region of possible velocities splits with formation of two disconnected regions; (d) plot for $u_{20}=2.2$ illustrates appearance of two disconnected regions of possible soliton velocities $V$.

Figure 5

Plot of the curve (41) (solid line) and of the straight lines $X-Y=U_0$ (dashed lines): (b) corresponds to a single region of possible values of $V$ [see Fig. 2]; (c) corresponds to the relative velocity at which the region splits into two regions [see Fig. 2]; (d) corresponds to two separated regions [see Fig. 2].

Figure 6

Plots of the densities of the components ${\rho }_1$ and ${\rho }_2$ as functions of $\xi$ [see Eqs. (47)]. The parameters are equal to ${\rho }_0=1.0,\gamma =2.5,V=0.4$.

Figure 7

Plots for the solution (56) of the Kowalevski equations for the values of the parameters $a=0.8,b=0.4,c=2$ which correspond to ${\nu }_1=-0.5,{\nu }_2=0.75,{\nu }_3=2.0,{\nu }_3=3.0,{\nu }_5=5.125$. The initial conditions are given by $q_1\left(0\right)={\lambda }_2,q_2\left(0\right)={\lambda }_3$. (a) Plots of $q_1$ and $q_2$ for the interval of $\xi$ corresponding to full cycle of $q_2$ variable; dashed lines indicate the values of ${\nu }_i,i=1,2,3,4$; (b) plots of the components densities ${\rho }_1$ and ${\rho }_2$ and of the total density $\rho ={\rho }_1+{\rho }_2$.