1. Introduction

Spatiotemporal solitons (STSs) in optical media have attracted much attention [1–8]. A spatiotemporal soliton is referred to as a “light bullet” localized in all spatial dimensions and in the time dimension. The generation of a “light bullet” might be of importance in soliton-based communication systems. An optical vortex soliton is a self-localized nonlinear wave, who has a point (“singularity”) of zero intensity, and with a phase that twists around that point, with a total phase accumulation of $2\pi S$ for a closed circuit around the singularity [9]. The quantity S is an integer number known as vorticity or topological charge of the solution. The collisions between solitons or vortices have been reported in conservative systems [10,11].

Complex Ginzburg-Landau (CGL) equations are well known as basic models of the pattern formation in various nonlinear dissipative media [12–17]. The CGL equation with the cubic-quintic (CQ) nonlinearity has been widely used in nonlinear dissipative optics, due to the clear physical meaning of all its terms in any particular application. Among the important application are passively mode-locked laser systems and optical transmission lines [18]. Stable fundamental spatiotemporal dissipative solitons (DSs) (S = 0) in optical models based on CQ CGL equation [19–22]. 3D vortex solitons with S = 1, 2, and 3 also have been obtained recently, as solutions to the CQ CGL equation [23–25]. Three alternative outcomes of collisions have been studied between 3D dissipative solitons or vortices [26–28]. However, the influence of phase between participating vortex solitons on the collisional outcomes has not been studied in detail. By modulating relative phase, the variation of the interaction should have an important impact on the collisional outcomes. The application of solitons in all-optical devices, given their particle-like properties in collision and interaction has been discussed extensively in conservative systems [29,30] and dissipative systems [31,32]. The features have potential applications in optical switching and logic gates based on interaction of optical solitons.

In this work, we investigate the dynamic of collisions between coaxial vortices with S = 0, 1, and 2 by introducing relative phase $\varphi$ in 3D CQ CGL model. By varying the collision momentum, three generic outcomes of collisions can be observed. The modulation of $\varphi$ on collision of vortices are systematically simulated. Moreover, the influence of cubic-gain coefficient on the dynamical regime of three collisional outcomes is considered.

2. The model

We consider the 3D CQ CGL equation in a general form [26–28]

Stationary localized solutions to Eq. (1) are sought for in the usual form:

A family of stable solitary-vortex solutions to Eq. (1) was constructed in Ref [26], by dint of direct simulations of the radial equation, which was obtained by the substitution of $u(z,x,y,t)=\psi (z,r,t)\exp (iS\theta ),$in Eq. (1) (with $\gamma =0$), i.e.,

Stationary solutions were found as attractors of this equation, generated by the evolution of input pulses $\Psi (0,r,t)=A{r}^{\left|S\right|}\exp [-\frac{1}{2}(\frac{r^2}{{\rho }^2}+\frac{t^2}{{\tau }^2})]$, with constants A, $\rho$, and $\tau$ [26].

Results of systematic simulations of soliton collisions may be adequately represented by outcomes observed at the following values of parameters in Eq. (1): D = 1 (anomalous GVD), $\mu =1$, $\epsilon =2.2$, $\nu =0.1$, $\delta =0.4$, $\beta =0.5$. In this case, the 3D vortices with S = 0, 1, and 2 are all stable, being characterized by the following values of the energy (alias norm):

To simulate the collisions between solitary vortices, at z = 0, a pair of stable solitary vortices are considered in the form of $\psi (r,t\text{+}\frac{t_0}{2})\exp (iS\theta )$ and $\psi (r,t\text{-}\frac{t_0}{2})\exp (iS\theta )$, with S = 0, 1, or 2, which are separated by large distance $t_0$. In this paper, we took $t_0\text{=}12$. The two vortices are set in motion in the t direction, i.e., multiplying each soliton by $\exp (\pm i\chi t)$. Moreover, the relative phase $\varphi$ between them also is introduced, the full initial configuration $u(0,r,t)$:

We have solved Eq. (1) using a split-step Fourier method with typical transverse and longitudinal step sizes $\Delta x=\Delta y=0.2$ and $\Delta z=0.1$ in all cases below. The second-order derivative terms in x and y are solved in Fourier space with the periodic boundary conditions. Other linear and nonlinear terms in the equation are solved in real space using a fourth-order Runge-Kutta method. We have studied collisions for DS pairs with coaxial vortices_(S, S) = (0, 0), (1, 1), and (2, 2). Gradually increasing initial kick $\chi$, three generic outcomes are identified With $\varphi =0$:

 

Fig. 1 Three outcomes of collision between two in-phase 3D DS pairs with coaxial vorticities ($S_1=1$,$S_2=1$). (a): merger of the two vortices at $\chi$ = 1. (b): add an extra coaxial vortex at $\chi$ = 2. (c): quasi-elastic passage through each other at $\chi$ = 2.5.

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In the following study, we are chiefly interested in the impact of relative phase $\varphi$ on the above three collisional outcomes.

3. Results and analysis

First, we numerically simulate the outcomes of collision by adding the relative phase $\varphi$ between two fundamental solitons ($S_1=0$,$S_2=0$). For $\chi <1.4$, there exists a critical value $\varphi \le {\varphi }_{\max }$ that the two solitons still fuse into one. But the merger-soliton acquires motion in t axis. Figure 2(a) shows the evolution of the merger of the two solitons with a motion in t axis for $\chi \text{=}1$ and $\varphi \text{=}0.1\pi$. Furthermore, the motion of merger-resulting soliton increases with growth of $\varphi$. For $\varphi >{\varphi }_{\max }$, the two solitons will bounce off each other with strong repulsive force [shown in Fig. 2(b)]. The relationship between ${\varphi }_{\max }$ and $\chi$ is shown in Fig. 2(f). Clearly, there is a significant range of $\varphi$ which can be used to modulate the motion of the merger soliton in t axis. The above feature leads to the merger-soliton with $\varphi \ne 0$ have more delay or advance in time axis than $\varphi \text{=}0$, which can be used for switching by controlling of phase difference.

 

Fig. 2 Relative phase controls outcomes of collision between two 3D fundamental solitons ($S_1=0$,$S_2=0$). (a): for $\varphi =0.1\pi$, merger of the two solitons into one acquires motion in t axis at $\chi$ = 1. (b): for $\varphi =0.\text{2}\pi \ge {\varphi }_{\max }$, solitons bounce off each other at $\chi$ = 1. (c): for $\varphi =0.2\pi$, the extra solitons acquires motion in t axis at $\chi$ = 1.7. (d): for $\varphi =0.3\pi \ge {\varphi }_{\max }$, (e): the relationship between ${\varphi }_{\max }$ and $\chi$ for the dynamics of merger. (f): the relationship between ${\varphi }_{\max }$ and $\chi$ for dynamics of adding extra soliton.

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For creation of an extra soliton in $1.4\le \chi \le 1.8$, we find there also exists a critical value ${\varphi }_{\max }$. For $\varphi \le {\varphi }_{\max }$, the two solitons, upon collision also create an additional one, whose motion can be modulated by $\varphi$. We show two typical examples in Fig. 2(c), the extra solitons have a motion along t axis for $\chi =1.7$ and $\varphi =0.3\pi$. Obviously, the motion of extra soliton increases with the growth of $\varphi$. Through this increase upon increase of $\varphi$, the velocity of the extra soliton gradually approaches that of the original solitons. If the critical value ${\varphi }_{\max }$ is exceeded, the extra soliton with large velocity interacts with one of original solitons [shown in Fig. 3(d)]. The extra soliton with $\varphi \ne 0$ have more delay or advance in time axis than in-phase. It also can be used for switching by modulation of phase difference.

 

Fig. 3 Relative phase controls outcomes of collision between two coaxial vortices with ($S_1=1$,$S_2=1$). (a): merger of the two vortices into one acquires motion in t axis for $\varphi =0.1\pi$, at $\chi$ = 1. (b): the extra vortex acquires motion in t axis for $\varphi =0.3\pi$, at $\chi$ = 2. (c): quasi-elastic passage through each other for $\varphi =0.3\pi$, at $\chi$ = 2.5. (d): the relationship between ${\varphi }_{\max }$ and $\chi$ for the dynamics of merger of the two vortices. (e): the relationship between ${\varphi }_{\max }$ and $\chi$ for the dynamics of adding extra vortex.

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Second, the similar control effect of relative phase $\varphi$ is studied on the collision between vortices with ($S_1=1$,$S_2=1$) by numerical simulation. The two vortices fuse into one In $\chi <1.7$. The merger vortex acquires motion in t axis by introducing $\varphi$. Figure 3(a) shows the evolution of collision for $\chi \text{=}1$ and $\varphi \text{=}0.1\pi$. The relationship between critical value ${\varphi }_{\max }$ and $\chi$ is shown in Fig. 3(d). The region of creation of an extra vortex is $1.7\le \chi \le 2.3$. The motion of extra vortex also can be effectively modulated by $\varphi$. A typical example is shown in Fig. 3(b), and the relationship between critical value ${\varphi }_{\max }$ and $\chi$ is shown in Fig. 3(e). On the other hand, for quasielastic interactions, in $\chi >2.3$, the two collisional vortices also pass through each other [shown in Fig. 3(c)]. By the comparison with Fig. 1(c), the variation of relative phase $\varphi$ cannot change the outcome of the collisions.

Third, the influence of variation of relative phase $\varphi$ on the collision between vortices with ($S_1=2$,$S_2=2$) is shown in Fig. 4. Figure 4(a) shows that the two vortices fuse into one ($\chi <1.7$), which acquires motion in t axis for $\chi \text{=}1$ and $\varphi \text{=}0.1\pi$. The relationship between critical value ${\varphi }_{\max }$ and $\chi$ is shown in Fig. 4(c). Figure 4(b) shows that the motion of extra vortex ($1.7\le \chi \le 1.9$) also can be modulated by $\varphi$, and the relationship between critical value ${\varphi }_{\max }$ and $\chi$ is shown in Fig. 4(d).

 

Fig. 4 Relative phase controls outcomes of collision between two coaxial vortices with ($S_1=2$,$S_2=2$). (a): merger of the two vortices into one acquires motion in t axis for $\varphi =0.1\pi$, at $\chi$ = 1. (b): the extra vortex acquires motion in t axis for $\varphi =0.3\pi$, at$\chi$ = 1.8. (c): the relationship between ${\varphi }_{\max }$ and $\chi$ for the dynamics of merger of the two vortices. (d): the relationship between ${\varphi }_{\max }$ and $\chi$ for the dynamics of adding extra vortex.

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Through above studies on the evolution of collision between dissipative vortices with S = 0, 1, and 2, we find that three second vortex pulses are generated after collision. The energy transfer from two bilateral pulses to middle vortex pulse with growth of $\chi$. The second vortex pulses with enough energy can self-trap into stable vortices; otherwise rapidly dissipate. So there are three generic collisional outcomes. At a small $\chi$, the middle vortex pulse self-trap into stable vortex, but the two bilateral vortex pulses without enough energy dissipate; for a larger $\chi$, the middle pulse without enough energy dissipates, but the two bilateral pulses can self-trap into stable vortices; for the intermediate region, all of three vortex pulses can self-trap into stable vortices. Furthermore, the relative phase between collisional vortices can effectively modulated the motion of middle pulse. So, the motion of merger vortex at a small $\chi$ and the extra vortex at an intermediate $\chi$ can be modulated by the relative phase $\varphi$.

In addition, we also study the influence on the collisional outcomes by the variety of cubic-gain $\epsilon$. The region of three collisional outcomes with S = 0, 1, and 2 is shown in Fig. 5(a), 5(b), and 5(c), respectively. For dissipative system of Eq. (1), linear loss $\delta$, quantic-loss $\mu$, and cubic-gain $\epsilon$ maintain energy balance. By increasing of gain coefficient (cubic-gain $\epsilon$) or reducing of loss ($\delta$ or $\mu$), Three second vortex pulses generated after collision can obtain more energy, which benefit in self-trap into stable vortices. So the dynamic region of creating an extra soliton (between the red circle and black square dotted line) gradually increase with the growth of cubic-gain $\epsilon$. We think that the similar effect can be achieved by reducing $\delta$ or $\mu$.

 

Fig. 5 (a), (b), and (c): The region of the three types of collisions by the variety of $\epsilon$ between two vortices with ($S_1=0$,$S_2=0$), ($S_1=1$,$S_2=1$), and ($S_1=2$,$S_2=2$).

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4. Conclusions

In summary, we have performed a systematic analysis of collisions between with stable 3D dissipative vortex solitons with S = 0, 1, and 2 in the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity. Depending on the initial kick applied to the vortices, $\pm \chi$, phase modulation on three generic outcomes of the collisions have been identified: at small $\chi$, there is an effectively region $\varphi \le {\varphi }_{\max }$ for controlling the velocity of merger of the two vortices into one; at intermediate $\chi$, there also exists a region $\varphi \le {\varphi }_{\max }$, with which the velocity of the extra vortex increases; at larger $\chi$, the relative phase lose efficacy, because the collisional vortices pass through each other in a quasielastic collision. The properties of phase modulation on collision have the potential application of enabling the design of optical switches and logic gates based on collision of solitons. In addition, upon reduction of cubic-gain coefficients ($\epsilon$), the intermediate dynamic regime for creation of an extra soliton will gradually decreases.