Strongly localized moving discrete dissipative breather-solitons in Kerr nonlinear media supported by intrinsic gain

We investigate the mobility of nonlinear localized modes in a generalized discrete Ginzburg-Landau-type model, describing a one-dimensional waveguide array in an active Kerr medium with intrinsic, saturable gain and damping. It is shown that exponentially localized, traveling discrete dissipative breather-solitons may exist as stable attractors supported only by intrinsic properties of the medium, i.e., in the absence of any external field or symmetry-breaking perturbations. Through an interplay by the gain and damping effects, the moving soliton may overcome the Peierls-Nabarro barrier, present in the corresponding conservative system, by self-induced time-periodic oscillations of its power (norm) and energy (Hamiltonian), yielding exponential decays to zero with different rates in the forward and backward directions. In certain parameter windows, bistability appears between fast modes with small oscillations and slower, large-oscillation modes. The velocities and the oscillation periods are typically related by lattice commensurability and exhibit period-doubling bifurcations to chaotically “walking” modes under parameter variations. If the model is augmented by intersite Kerr nonlinearity, thereby reducing the Peierls-Nabarro barrier of the conservative system, the existence regime for moving solitons increases considerably, and a richer scenario appears including Hopf bifurcations to incommensurately moving solutions and phase-locking intervals. Stable moving breathers also survive in the presence of weak disorder.

Figure 1

Main figure: velocity $v$ vs gain parameter $g$ for moving localized solutions of Eq. (1) with $C=1$. Other parameters are $V_n=Q=0$ and $f_d$ given by relations (2) with (9). Horizontal dashed lines indicate the velocity gap. Inset: Power (4) vs time for the bistable fast (solid [red], small oscillations) and slow (dashed [blue], large oscillations) solutions having the same $g=2.095$.

Figure 2

Snap shots of intensity $A_n$ in logarithmic scale for two right-moving solutions with $g=2.07$ (right peak [red], fast solution) and $g=2.18$ (left peak [blue], slow solution), respectively. Other parameters are the same as in Fig. 1.

Figure 3

Maximum intensity $A_n^{\left(\max \right)}$ versus gain parameter $g$ for different lattice sites, illustrating the locking between internal oscillations and lattice translations for steadily moving solutions, and the period-doubling scenario in the upper end of the existence regime. Parameter values are the same as in Fig. 1.

Figure 4

Dynamics of a moving soliton with velocity $v\approx 0.052$ in the period-3 window [$(p,q)=(1,3)$] for $C=0.46$ and $g=2.0574$ (other parameters same as in Fig. 1). Upper figure: intensity $A_n$ vs time for six consecutive sites (larger $n$ corresponds to later peak time). Note the identical curve shapes for every third site (the horizontal line at 0.6 is a guide to the eye). Lower figure: the corresponding oscillations of the quantities $P$ (4) (solid [red] line) and $H$ (5) (dashed [blue] line).

Figure 5

Main figure: velocity $v$ vs gain parameter $g$ for moving localized solutions of Eq. (1) with $C=1$, $Q=0.4$, and other parameters same as in Fig. 1. Inset: velocity vs coupling constant $C$ for fixed gain parameter $g=2.2$ and other parameters the same as in the main figure.

Figure 6

Dynamics of a moving “slow” soliton for $Q=0.264$, $C=0.3$, and $g=2.2$ (other parameters same as in Fig. 1). Upper figure: intensity $A_n$ vs time for 23 consecutive sites (larger $n$ corresponds to later peak time). Notice that while peaks are equidistant, corresponding to a well-defined velocity $v\approx 0.285$, the modulation of the peaks is apparently incommensurate; i.e., no two peaks corresponding to different sites have identical shape. Lower figure: Fourier spectrum of the corresponding oscillations of the power $P$ (4), calculated over 40 000 time units. The main peak at $\omega =2\pi v\approx 1.793$ corresponds to the translational motion, while the first peak at $\omega \approx 0.231$ corresponds to the modulation. All other peaks seen are linear combinations of these.

Figure 7

Dynamics of a soliton moving with velocity $v\approx 0.205$ in the $(p,q)=(2,5)$ phase-locked regime for $Q=0.2285$ (other parameters same as in Fig. 6). Upper figure: intensity $A_n$ vs time. Note that the pattern periodically repeats itself after translating $q=5$ sites. Lower figure: Fourier spectrum of the oscillations of the power $P$ (4), calculated over 40 000 time units. The peak at ${\omega }_1=2\pi v\approx 1.286$ corresponds to the translational motion, while the second peak at ${\omega }_2\approx 0.514$ corresponds to the main modulation. The first peak is ${\omega }_1-2{\omega }_2={\omega }_2/2$, and all other peaks seen are multiples of this frequency.

Figure 8

Dynamics of a soliton moving with velocity $v\approx 0.977$ in a lattice with uniform disorder of strength $V_0=0.1$, for $g=2.06$ and other parameters same as in Fig. 1. The soliton appears as an attractor using the soliton for $V_0=0$ as initial condition. Upper figure: intensity $A_n$ vs time. Only a part of the dynamics is shown for a lattice with 2405 sites and periodic boundary conditions. Lower figure: the corresponding oscillations of the quantities $P$ (4) (lower [blue] curve) and $H$ (5) (upper [red] curve). Note that after an initial transient, the patterns periodically repeat themselves after 2462 time units, corresponding to one round trip in the lattice.