Discrete dark solitons with multiple holes

We consider staggered dark solitons admitted by the discrete nonlinear Schrödinger equation with focusing cubic nonlinearity. In particular, we focus on the study of dark solitons with several holes characterized by the number of zeros in the uncoupled case. Such structures reveal interesting behaviors, such as stable intersite dark solitons. All of the structures have no counterpart in the strong coupling limit since they disappear in a saddle-node bifurcation. We also consider the evolution of structures with multiple holes representing an interaction between multiple dark solitons in a very discrete case.

Figure 1

Real part of the unstable eigenvalues of an on-site dark soliton of Eq. (1) as a function of the coupling parameter $C$.

Figure 2

(a) The oscillation amplitude of the phase field ${\psi }_n$ at each site and the value ${\mid {\psi }_n\mid }^2$ of an intersite dark soliton. The two holes move with out-of-phase configuration. (b) Real part of the unstable eigenvalues of the solution as a function of the coupling parameter $C$. The inset enlarges the lower branch where it becomes clear that the intersite-like dark soliton is stable for small $C$. (c) The evolution of an $A$ mode soliton at $C=0.2$. In the contour plot, lines correspond to intensities ${\mid {\psi }_n\mid }^2=0.5$, 1 and 1.5. The inset shows the detailed dynamics of a few sites at and around the holes. One can see that the holes also move in time in out-of-phase configuration.

Figure 3

A diagram of the existence and the stability region of dark solitons with multiple zeros in the $A$-mode configuration (–∗–) and the $B$-mode configuration (–엯–). The calculations for the $A$ modes [$B$ modes] are done with $N=201$ or $N=202$ for an odd [even] or even [odd] number of zeros, respectively.

Figure 4

The same as Fig. 2, but the two holes are in the $B$ mode configuration.

Figure 5

(a) A solution representing two dark solitons. The two solitons are only separated by one lattice site. (b) The evolution of a solution given in (a) at $C=0.2$. The inset shows the evolution of several sites at and around the holes.