Crossover dynamics of dispersive shocks in Bose-Einstein condensates characterized by two- and three-body interactions

We show that the perturbative nonlinearity associated with three-atom interactions, competing with standard two-body repulsive interactions, can change dramatically the evolution of one-dimensional (1D) dispersive shock waves in a Bose-Einstein condensate. In particular, we prove the existence of a rich crossover dynamics, ranging from the formation of multiple shocks regularized by nonlinear oscillations culminating in coexisting dark and antidark matter waves to 1D-soliton collapse. For a given scattering length, all these different regimes can be accessed by varying the density of atoms in the condensate.

Figure 1

Dispersionless evolution for ${\rho }_0=1.5$ as obtained from Eqs. (5, 6, 7) with (a) $\alpha =0.15$ ($\alpha <{\alpha }_c$) and (b) $\alpha =0.3$ ($\alpha >{\alpha }_c$) Numerical results from Eqs. (2) comparing the case (c) $\alpha =0.4$ with the case (d) of a standard GPE ($\alpha =0$) with positive cubic nonlinearity (i.e., attractive interactions).

Figure 2

Global phase diagram of shock dynamics in parameter plane $({\rho }_0\equiv {\rho }_0^{-},\alpha )$ [${\rho }_0^{+}=1$].

Figure 3

(a) Velocity of DSW edges ${\zeta }_t$, ${\zeta }_l$, and rarefaction edges ${\zeta }_{e-}$, ${\zeta }_{i-}$ vs $\alpha$; Color level plot of density evolution $\rho (x,t)$ from Eq. (1) for $\epsilon =0.01$ and (b) $\alpha =0$, (c) $\alpha =0.1$. For comparison, the edges of the DSW $x_l$, $x_t$ from Eq. (12), $x_{e-}$, $x_{i-}$ of the rarefaction wave from Eqs. (7) and (8) are reported. (d) Snapshots $\rho (x,t=8)$ from (b) and (c) showing the DSW modulated cnoidal wave for $\alpha =0,0.1$.

Figure 4

(a) Snapshots of density $\rho \left(x\right)$ obtained from Eq. (1) with $\alpha =0.3$, $\epsilon =0.01$, and a input jump ${\rho }_0=1.5$. Panels (b) and (c) present a zoom of the boxed evolution in (a), showing the opposite type of regularization taking place in the dynamics.

Figure 5

$(\alpha ,v)$ Stability map of the antidark soliton solutions of the CQNLS for ${\rho }_0^{-}=1.5$, with $v$ being the soliton velocity in the $(t,x)$ plane. The position of the antidark soliton of Fig. 4b is indicated as a point in the figure.

Figure 6

Snapshots of density $\rho$ vs $x$ from Eq. (1) for $\alpha =0.4$ and $\epsilon =0.01$: (a) shock regularization, (b) antidark collapse instability.