Vector solitons with an embedded domain wall

We present a class of soliton solutions to a system of two coupled nonlinear Schrödinger equations, with an intrinsic domain wall (DW) which separates regions occupied by two different fields. The model describes a binary mixture of two Bose-Einstein condensates (BECs) with interspecies repulsion. For the attractive or repulsive interactions inside each species, we find solutions which are bright or dark solitons in each component, while for the opposite signs of the intraspecies interaction, a bright-dark soliton pair is found (each time, with the intrinsic DW). These solutions can arise in the context of discrete lattices, and most of them can be supported in continuum settings by an external parabolic trap. The stability of the solitons with intrinsic DWs is examined, and the evolution of unstable ones is analyzed. We also briefly discuss the possibility of generating such families of solutions in the presence of linear coupling between the components, and an application of the model to bimodal light propagation in nonlinear optics.

Figure 1

Discrete bright-bright soliton pair separated by a domain wall. The top panel shows the norm of the solution, $P$ [defined as per Eq. (2)], as a function of $C$; the second panel shows a spatial profile of the two components for $C=1.85$, where the solution becomes unstable due to the presence of a real eigenvalue pair (additional unstable pairs appear at larger $C$). The third panel shows the spectral plane $({\lambda }_r,{\lambda }_i)$ for the eigenvalues, $\lambda ={\lambda }_r+i{\lambda }_i$. The bottom panel shows the principal eigenvalues bifurcating for $C\approx 0.655$ from the continuous spectrum and approaching the origin of the spectral plane. They become unstable at $C\approx 1.85$ (the second pair becomes real at $C\approx 1.86$), destabilizing the solution. The circles and triangles show the eigenvalues when they are still imaginary (stable), while the stars and crosses show them when they are real, i.e., unstable.

Figure 2

(Color online) The evolution of an unstable pair of discrete bright solitons separated by the domain wall is shown for $C=1.86$ by means of space-time contour plots of the absolute values of the wave functions, $\mid {\psi }_{1,n}\mid$ and $\mid {\psi }_{2,n}\mid$ (top and bottom panels). The solution is perturbed by its most unstable eigenmode.

Figure 3

(Color online) Discrete dark-dark soliton pair separated by a domain wall. The top three panels are similar to the respective ones in Fig. 1; however, the second and third panels correspond to $C=1$. Notice the two unstable quartets in the third panel (their precise location depends on the number of points used in the discretization; for these panels, we have used a 200-site lattice). The last two panels show (through the space-time contour plots of $\mid {\psi }_{1,n}\mid$ and $\mid {\psi }_{2,n}\mid$) the evolution of the unstable solution for $C=1$. The solution is perturbed by its most unstable eigenmode.

Figure 4

(Color online) Discrete bright-dark soliton pair separated by a domain wall. The coupling parameter value for this example is $C=0.07$, i.e., slightly below the stability border of the dark solitary wave. The second panel illustrates the stability of the configuration by showing the spectral plane of eigenvalues. The last two panels depict the evolution of an unstable bright-dark soliton for $C=0.08$.

Figure 5

Continuous bright-bright soliton pair separated by a domain wall. The top panel shows the solution’s norm $P$ [defined as per Eq. (4)] as a function of $\Omega$. The second panel shows the spatial profile of the solution (for the two components in solid and dashed line, respectively; the external potential is shown by dash-dotted line) for $\Omega =0.3$. The third panel shows the stability of this solution through the corresponding eigenvalues for $\Omega =0.3$ (no real pair). The bottom panel of the figure illustrates the result of continuation for the separation $\Delta$ between the pulses of this stable branch of solutions, as a function of the magnetic trap strength $\Omega$. The numerical result is given by the solid line while the variational prediction (see text and Appendix ) is shown by the dashed line.

Figure 6

(Color online) Continuous dark-dark soliton pair with an embedded domain wall. The top panel shows the continuation of the solution’s norm vs the trap’s strength $\Omega$. For the next panels, the four subplots show the profiles (top) and stability (bottom) of the solution for ${\Omega }^2=0.004$ (left) and ${\Omega }^2=0.04$ (right). The last two panels show, respectively, the separation between centers of mass of the two components, $\Delta$, and the real part of the most unstable eigenvalue of the configuration, ${\lambda }_r$, vs $\Omega$. It is clearly observed that the two dark solitons merge for $\Omega >0.25$. The solution is stable in two distinct intervals: $0.09<\Omega <0.155$ and $\Omega >0.195$.

Figure 7

(Color online) The two panels show the space-time contour plot of the square modulus of the two fields for $g_{12}=2=2g_{11}=2g_{22}$ and for ${\Omega }^2=0.004$.

Figure 8

(Color online) Continuous bright-bright soliton pair bound by linear coupling. The top panel shows the norm of the solution as a function of the coefficient $g_{12}$ which accounts for the nonlinear repulsion (the parameter controlling the linear coupling is fixed to $\kappa =1$). The dash-dotted line shows the branch with zero separation between the solitons $(\Delta =0)$. The two branches merge for $g_{12}⩽1∕3$. The second panel shows an example of the bright-soliton pair with the domain wall between them, for $g_{12}=1.1$. The third panel shows the dependence of the most unstable (real) eigenvalues for the $\Delta \ne 0$ (solid line) and the $\Delta =0$ (dash-dotted line) solutions. The bottom panel shows the pulse separation between the solitons in the bound-pair solution as a function of $g_{12}$. The solid and dashed lines, respectively, depict the numerical results and prediction of the variational approximation (see Appendix ).

Figure 9

(Color online) Evolution of a bright-soliton pair in the model with the linear coupling, for $g_{12}=1.1$ and $\kappa =1$. The solution is perturbed only by numerical noise, which is sufficient to provoke its very rapid transformation into a breathing pulse in one component, while the other one keeps practically no matter. The top panel shows the norm of each component as a function of time. The middle and lower panels show the evolution in terms of space-time contour plots of the square modulus of the two fields.

Figure 10

Branch of the stable asymmetric bright soliton pair in the presence of linear coupling. The top panel depicts the norm of the bright solitons in each component as $g_{12}$ is varied. This branch is found to be stable in all of its domain. The middle and bottom panels show an example of such a configuration and its stability spectrum for $g_{12}=1$. The linear coupling for these results is fixed at $\kappa =1$.

Figure 11

(Color online) Weakly unstable asymmetric bright-soliton pair for $g_{12}=1.1$ and $\kappa =1$. The top panel is the solution profile, and the second panel illustrates its (in)stability. The third panel shows the evolution of the norms of the two components as a function of time. The two bottom panels show space-time contour plots of square modulus of the two fields.