Exact numerical solutions for dark waves on the discrete nonlinear Schrödinger equation

In this paper we study numerically existence and stability of exact dark waves on the (nonintegrable) discrete nonlinear Schrödinger equation for a finite one-dimensional lattice. These are solutions that bifurcate from stationary dark modes with constant background intensity and zero intensity at a site, and whose initial state translates exactly one site each period of the internal oscillations. We show that exact dark waves are characterized by an oscillatory background whose wavelength is closely related with the velocity. Faster dark waves require smaller wavelengths. For slow enough velocity dark waves are linearly stable, but when trying to continue numerically a solution towards higher velocities bifurcations appear, due to rearrangements in the oscillatory tail in order to make possible a decreasing of the wavelength. However, in principle, one might control the stability of an exact dark wave adjusting a phase factor which plays the role of a discreteness parameter. In addition, we also study the regimes of existence and stability for stationary discrete gray modes, which are exact solutions with phase-twisted constant-amplitude background and nonzero minimum intensity. Also such solutions develop envelope oscillations on top of the homogeneous background when continued into moving phase-twisted solutions.

Figure 1

Stationary dark breather for ${\Omega }^{\prime }=\pi ∕3$, ${\omega }_0=4$ and $C=1$.

Figure 2

Oscillatory instabilities of the stationary dark breathers vs the period for different lattice sizes and parameter values $C=1$, ${\omega }_0=4$.

Figure 3

Time evolution of an exact dark wave. The initial state translates one site each period $T$ of the internal oscillations.

Figure 4

(a) Snapshots of exact moving dark breathers corresponding to $T=4$, $T=4.5$, and $T=6$ from top to bottom. Note how the intensity of the background decreases when $T$ increases. Other parameters values: $C=1$, ${\omega }_0=4$. (b) $\mathrm{Re}\left[{\psi }_n\left(0\right)\right]$ (full circles), $\mathrm{Im}\left[{\psi }_n\left(0\right)\right]$ (open circles), and ${\mid {\psi }_n\left(0\right)\mid }^2$ (stars) for an exact moving solution with $T=4$.

Figure 5

Instabilities found along a numerical continuation towards lower values of $T$. Shortly after the harmonic bifurcations the continuation path stops. At this point we have to jump into another family of solutions increasing the continuation step. For $T<7$ we begin to observe oscillatory instabilities which become stronger. Other parameter values: $C=1$, ${\omega }_0=4$, $N=61$.

Figure 6

Exact mobile dark breathers just at the first bifurcation $T_1=8.456$ (pluses) and for $T=8.3$ (circles) after jumping over the bifurcation. The full circles correspond to the continuation of this last solution towards the bifurcation point. Other parameter values: $C=1$, ${\omega }_0=4$.

Figure 7

(a) When we reverse the continuation path after jumping over a bifurcation, an avoided collision between a pair of conjugated Floquet eigenvalues gives raise to huge changes in the oscillatory tail of the solutions. (b) These rearrangements enclose oscillatory instabilities.

Figure 8

Exact mobile dark breathers just at the bifurcation $T_5=6.029$ (circles) and for $T=5.6$ (squares) after jumping over the bifurcation. Other parameter values: $C=1$, ${\omega }_0=4$.

Figure 9

Inversion of the continuation path after jumping over the fifth bifurcation. When we approach the bifurcation point the smooth oscillating tails are destroyed.

Figure 10

(a) An avoided collision between a pair of conjugated Floquet eigenvalues points out the excitation of the extended mode shown in Fig. 11. (b) This excitation gives raise to numerous oscillatory instabilities.

Figure 11

Projection of Fig. 9 on to the $(j,{\mid {\psi }_j\mid }^2)$ plane. Notice the excitation of an extended mode with wave number close to $\pi$.

Figure 12

Spatial oscillation period $2\pi ∕q$ vs $T$ for small-amplitude tail oscillations of the form (10) fulfilling (7), as obtained from (11). Solid (dashed) lines correspond to a positive (negative) sign in (11), with, from top to bottom, $n$ ranging from $-1$ to $-7$ (1 to 6). As in previous figures, $C=1$ and ${\omega }_0=4$.

Figure 13

Exact dark waves for $T=12$ and three different values of ${\omega }_0$: 4.0 (circles), 3.64 (full circles), and 4.25 (pluses). $\mathrm{C}=1$.

Figure 14

Instabilities vs the phase ${\omega }_0$ for a fixed value of the period $T=8.4$.

Figure 15

Instabilities vs the phase ${\omega }_0$ for a fixed value of the period $T=5$.

Figure 16

Stationary discrete gray breather in DNLS, corresponding to a background wave with $Q=0.98\pi$. (a) shows, for $C=0.075$, intensity ${\mid {\psi }_j\mid }^2$ (minimum intensity is ${\mid {\psi }_{31}\mid }^2\approx 0.0052$); (b) shows nearest-neighbor phase shifts ${\alpha }_{j+1}-{\alpha }_j$ (with notation ${\psi }_j=\mid {\psi }_j\mid e^{i{\alpha }_j}$) (note that the total phase shift across the central site ${\alpha }_{32}-{\alpha }_{30}$ is not $\pi$ as it would be for a black breather). Lower figures with stability eigenvalues show that the solution is close to the threshold of Krein instability: $C=0.075$ is stable (c) while $C=0.076$ is unstable (d).

Figure 17

(a) Continuation of stationary discrete gray soliton from continuous limit $C=C_{\mathit{max}}$ [from (14)] to anticontinuous limit $C=0$ at fixed $Q=0.6\pi$. The smallest minimum intensity ${\mid {\psi }_{\mathit{min}}\mid }^2\simeq 0.9398$ is obtained for $C\simeq 0.09$. Only the central part of a larger chain is shown. (b) shows the real part of the stability eigenvalues during the continuation. The unstable eigenvalues are complex for $C\gtrsim 0.06$ and real for $C\lesssim 0.06$.

Figure 18

(a) Solid line: Location of the bifurcation interrupting a monotonic continuation at fixed $Q$ of a continuumlike gray soliton towards the anticontinuous limit $C=0$. Dashed line: Location of the Krein collision causing oscillatory instability. A stable stationary discrete gray soliton exists only in the area between the lines (lower right part of the figure). (b) Patterns for the central parts of two simultaneously existing solutions at $Q=0.95\pi$, $C=0.13$. (+): Solution continued from continuous limit; (×): solution continued from anticontinuous limit.