Solitary matter wave in spin-orbit-coupled Bose-Einstein condensates with helicoidal gauge potential

We analytically and numerically study the different types of solitary wave in the two-component helicoidal spin-orbit coupled Bose-Einstein condensates (BECs). Adopting the multiscale perturbation method, we derive the analytical bright and dark solitary wave solutions of the system, and the stationary and moving bright (dark) solitary waves are obtained. The effects of spin-orbit coupling, the helicoidal gauge potential, the momentum, the Zeeman splitting, and the atomic interactions on the solitary wave types are discussed, and it is found that the coupling of these physical parameters can manipulate different types of solitary waves in the system. The results indicate that the helicoidal gauge potential breaks the symmetric properties of the energy band of the system and adjusts the energy band structure, thus further effecting the solitary wave properties, i.e., stationary or moving solitary wave, bright, or dark solitary wave. Correspondingly, the analytical predictions for exciting stationary or moving bright (dark) solitary wave in parameter space are obtained. In particular, the helicoidal gauge potential changes the solitary wave types drastically for the weak spin-orbit coupling, i.e., in the absence of the helicoidal gauge potential, only dark (bright) solitary wave solutions exist in the system with repulsive (attractive) atomic interaction; however, in the presence of the helicoidal gauge potential, both dark and bright solitary waves can exist in the system regardless of whether the atomic interaction is repulsive or attractive. In addition, we investigate the stability of solitary waves and obtain the stability regions of different types of solitary waves by applying the linear stability analysis. The dynamic evolution results of the solitary waves by the direct numerical simulation not only validate the linear stability analysis but also confirm the analytical prediction of the solitary waves. Finally, the collision effects between solitary waves are also presented by the numerical simulation. It is shown that the interactions between solitary waves in the system have both elastic and inelastic collisions, which are closely related to the position of solitary wave states in the linear energy band. Our results provide a potential way to adjust the types of solitary waves in BECs with helicoidal gauge potential.

Figure 1

The lower branch of the linear energy spectrum: (a) for different Zeeman splitting $\mathrm{\Delta }$ with $\beta =0$ and (b) for different helicoidal gauge potential $\beta$ with $\mathrm{\Delta }=6$. Here $\alpha =1.5$.

Figure 2

Distribution of (a) group velocity $v$ and (b) dispersion coefficient $P$ of different helical gauge potential $\beta$ in momentum $k$ space. The other parameters $\alpha =1.5, \mathrm{\Delta }=6$.

Figure 3

The group velocity region (i.e., $v<0, v=0, v>0$) for different $k$ in $\alpha -\beta$ plane. Here $\mathrm{\Delta }=6$.

Figure 4

Bright and dark solitary wave regions for different $k$ in $\alpha -\beta$ plane. (a) $S>0$ with $g_1=g_2=g_{12}=1$. (b) $S<0$ with $g_1=g_2=g_{12}=-1$. Here $\mathrm{\Delta }=6$.

Figure 5

The contour plots show the evolution of the total density for stationary solitary waves, bright solitary waves for $g_1=g_2=g_{12}=-1, \kappa =0$ in (a1) and (a2) and dark solitary waves for $g_1=g_2=g_{12}=1, \theta =0$ in (b1) and (b2). The other parameters $\alpha =1.5, \beta =3, \mathrm{\Delta }=6$, and $k=-3.34$. [(a) and (b)] Maximum growth rates of perturbation as functions of the chemical potential for the stationary bright solitary wave and dark solitary wave, respectively. The insets in (a) and (b) are the density profiles of solitary waves at $t=0$.

Figure 6

The contour plots show the evolution of the total density for moving solitary waves, bright solitary waves for $\beta =1.2, \kappa =1$ in (a1) and (a2), dark solitary waves for $\beta =-2, \theta =\pi /10$ in (b1) and (b2). The other parameters $\alpha =1.5, g_1=g_2=g_{12}=1, \mathrm{\Delta }=6$ and $k=1.5$. [(a) and (b)] Maximum growth rates of perturbation as functions of the chemical potential for the moving bright solitary wave and dark solitary wave respectively. The insets in (a) and (b) are the density profiles of solitary waves at $t=0$.

Figure 7

The top row: The contour plots of the evolution of the total density for the moving solitary waves. The bottom row: Head-on collisions of the solitary waves which corresponds to the top row in the $(x,t)$ plane, where $\kappa =3$ ($\kappa =-3$) for all left-moving (right-moving) solitary waves. The insets in the bottom row are the schematic diagrams of the position of solitary wave states in the linear energy band. The other parameters $\alpha =1.5, \mathrm{\Delta }=6, g_1=g_2=g_{12}=-1$, and ${\epsilon }^2{\omega }_0=-0.3$.