Interactions of the solitons in periodic driven-dissipative systems supporting quasibound states in the continuum

The paper is devoted to the dynamics of dissipative gap solitons in the periodically corrugated optical waveguides whose spectrum of linear excitations contains a mode that can be referred to as a quasi-bound state in the continuum. These systems can support a large variety of stable bright and dark dissipative solitons that can interact with each other and with the inhomogeneities of the pump. One of the focus points of this work is the influence of slow variations of the pump on the behavior of the solitons. It is shown that for the fixed sets of parameters the effect of pump inhomogeneities on the solitons is not the same for the solitons of different kinds. The second main goal of the paper is systematic study of the interaction between the solitons of the same or different kinds. It is demonstrated that various scenarios of intersoliton interactions can occur: The solitons can repulse each other or get attracted. In the latter case, the solitons can annihilate, fuse in a single soliton, or form a new bound state depending on the kinds of the interacting solitons and on the system parameters.

Figure 1

Bifurcation diagram of the uniform states (thin blue line is for symmetric states and bold green line is for hybrid states) and bright solitons (black line), where $W=|U_b{|}^2+|U_d{|}^2=|U_{+}{|}^2+{\left|U_{-}\right|}^2$ is the intensity of uniform states or maximum intensity of the solitons; dynamically unstable solutions are shown by the dashed line. The SSB points show where the spontaneous-symmetry-breaking bifurcation takes place; the red (shaded) rectangle stresses unstable uniform hybrid states and solitons; the thin black vertical lines mark the region of existence of resting and walking dark solitons. The inset schematically shows the system under consideration. The parameters are $\alpha =-1$, $\delta =0.1155$, $\sigma =1$, $\gamma =0.001$, and $\mathrm{\Gamma }=0.299$.

Figure 2

Space-time diagrams demonstrating the motion of the solitons caused by spatially nonuniform pumping. The colormap is used to show the intensity of the field $W(x,t)$ (yellow corresponds to high field intensity and blue is for low intensity). Panels (i) show modified (a) and (b) phase and (c) and (d) amplitude of the pumping field and panels (ii) show the final field distributions. All parameters are the same as in Fig. 1. (a) Sink and source dark solitons moving under the pumping field with modified phase $P=P_0{e}^{\mathrm{\Theta }\left(x\right)}$. (b) Bright soliton on the symmetric background set in motion by the pumping field with a nonuniform phase. (c) Amplitude gradient of the pumping field $P=P_0\left(x\right)$ forcing sink and source dark solitons to move. (d) Bright soliton moving under the pumping field with nonuniform amplitude.

Figure 3

Velocity dependence of the walking soliton at pumping amplitude. (a) The blue (middle) line shows velocity dependence in the uniform pumping field $P\left(x\right)=P_0$, the violet (bottom) one is for the pump field of the form $P\left(x\right)=P_0{e}^{ikx}$, and the green (top) one is for $P\left(x\right)=P_0{e}^{-ikx}$, with $k={10}^{-3}$. (b) The blue (middle) line is the same as in (a), the violet (top) curve shows velocity dependence on the pump of the form $P\left(x\right)=P_0+\mu x$, and the green (bottom) one is for $P\left(x\right)=P_0-\mu x$, with $\mu =5\times {10}^{-4}$.

Figure 4

Space-time diagrams demonstrating interactions of dark solitons. Panels (i) and (ii) show initial and final fields distributions, respectively. (a) Interaction of initially resting dark solitons resulting in walking soliton formation (see the inset) and further interaction of the walking soliton with the resting dark one, with $P=0.3$. (b) Interaction of dark solitons resulting in uniform hybrid state formation, with $P=0.54$. (c) Interaction of dark solitons resulting in uniform symmetric state formation, with $\delta =0.15$ and $P=0.3$. (d) Interaction of two walking solitons resulting in the formation of new walking soliton with a larger width, with $P=0.4$. The other parameters are the same as in Fig. 1.

Figure 5

Space-time diagrams demonstrating interactions of solitons. The colormap is used to show (a) and (b) the intensity of the field $W(x,t)$ and (c) and (d) the amplitude $\sqrt{W(x,t)}$. Detuning for (a) and (b) is $\delta =0.05$; the other parameters are the same as in Fig. 1. Panels (i) and (ii) show initial and final field distributions, respectively. (a) Two bright solitons at the symmetric backgrounds collide and form a single soliton. The initial distance between solitons is $l=7.9$. (b) Two bright solitons repulse each other with initial distance $l=9.4$. In (a) and (b) $P=0.8$. (c) Repulsive interaction of a dark source soliton and a bright source soliton, with $P=0.5244$. (d) Interaction of a walking soliton with a bright one, resulting in formation of the moving bound state, with $P=0.51$.