## Abstract

We investigate collisions of quasi-one-dimensional dissipative solitons (DSs) for a large class of initial conditions, which are not temporally asymptotic quasi-one-dimensional DSs. For the case of sufficiently small approach velocity and sufficiently large values of the dissipative cross-coupling between the counter-propagating DSs, we find non-unique results for the outcome of collisions. We demonstrate that these non-unique results are intrinsically related to a modulation instability along the crest of the quasi-one-dimensional objects. As a model, we use coupled cubic–quintic complex Ginzburg–Landau equations. Among the final results found are stationary and oscillatory compound states as well as more complex assemblies consisting of quasi-one-dimensional and localized states. We analyse to what extent the final results can be described by the solutions of one cubic–quintic complex Ginzburg–Landau equation with effective parameters.

## 1. Introduction

Quasi-one-dimensional objects and their interactions have been studied in different contexts. In hydrodynamics, a fluid layer heated locally by a nearly one-dimensional heater, and subjected to horizontal and vertical temperature gradients shows quasi-one-dimensional convective rolls and quasi-one-dimensional stationary and time-dependent cellular patterns [1,2]. In chemical systems, interaction of counter-propagating concentration waves on a Pt(110) surface has been studied in reference [3]. These quasi-one-dimensional objects exhibit interpenetration and partial annihilation after collision. Recently, in the context of biological physics, it has been reported that an arc-shaped multicellular structure of non-chemotactic mutants presents solitonic behaviour [4], which has been modelled using a particle-based simulation [5].

The topic under study in this article is the collision of quasi-one-dimensional dissipative solitons. Quasi-one-dimensional solutions are localized in one spatial dimension and spatially extended in the other. By dissipative solitons, we mean solutions whose existence is based on the balance of nonlinearity, dispersion, gain and loss [6]. Dissipative solitons have been studied either experimentally, in systems such as binary fluid convection [7–9], surface reactions [3,10], granular matter [11,12], starch suspensions [13,14], nonlinear optics [15–17] and sheared electroconvection in liquid crystals [18], or theoretically by means of prototype equations, such as envelope equations of the Ginzburg–Landau-type [19–40], order parameter equations [41–43] or reaction–diffusion equations [44–49]. The studies on localized solutions in Ginzburg–Landau-type equations include studies in one [19–21,23,24,26–29,33,34,36–39] as well as in two [19,22,23,25,26,30,31,35,36,40] spatial dimensions. We note that reference [49] discusses a combination of a simple reaction–diffusion system with a cubic–quintic complex Ginzburg–Landau equation with possible application to living systems. Stable quasi-one-dimensional dissipative solitons have been first observed in an envelope equation applicable to nonlinear optics [23] and later in order parameter equations [30,31,41,43]. Experimentally, collisions of dissipative solitons and holes have been investigated leading to interpenetration [10], mutual annihilation [14], partial annihilation [3,9,14] or compound states [9]. Using modelling, the collision of dissipative solitons has been studied for various systems leading to interpenetration [20,22,25,26,33,34,36,42,45], complete annihilation [20–22,25,26,36,45], partial annihilation [21,26,27,36] and compound states [22,27,33,34,36,40,42]. Collision of counter-propagating waves leading to domain walls has been studied experimentally [50] and theoretically [51] for binary fluid convection.

Recently, we have reported the coexistence of quasi-one-dimensional and quasi-two-dimensional azimuthally symmetric stationary and exploding pulses in the two-dimensional cubic–quintic complex Ginzburg–Landau equation [40]. In a previous article, we have investigated in detail the outcomes of the collisions of counter-propagating quasi-one-dimensional solutions starting in their asymptotic state. We observed interpenetration, one compound state and two compound states as possible results. The latter is a consequence of a quasi-linear instability induced by the cross-coupling between the equations [52].

Here, we report non-unique outcomes of collisions of quasi-one-dimensional dissipative solitons which are not in their asymptotic state. The paper is organized as follows. In §2, we present the model and the numerical method used. In §3, we describe the non-unique results of collisions of quasi-one-dimensional objects followed, in §4, by an intuitive picture explaining the occurrence of oscillatory DSs and in §5 by conclusions.

## 2. Coupled complex subcritical cubic–quintic Ginzburg–Landau equations

In this article, we show the results of collisions of counter-propagating quasi-one-dimensional solutions, which have not reached their asymptotic regime, as it is shown in figure 2*a*. The system consisting of two coupled complex subcritical cubic–quintic Ginzburg–Landau equations reads
2.1and
2.2where the complex fields *A*(*x*,*y*,*t*) and *B*(*x*,*y*,*t*) vary slowly in time and space. is the two-dimensional Laplacian.

We have carried out our numerical studies for the following values of the parameters: *μ*=−0.9,*β*_{r}=1,*β*_{i}=0.8,*γ*_{r}=−0.1,*γ*_{i}=−0.6,*D*_{r}=0.125,*D*_{i}=0.5 and *c*_{i}=0. These parameters guarantee the existence of stationary stable quasi-one-dimensional solutions as initial conditions (without considering the interaction between the two equations) [40]. More negative values of *μ* lead to breathing quasi-one-dimensional solutions or to collapse. The existence of the quintic nonlinearities and at least one of the parameters *β*_{i},*γ*_{i} or *D*_{i} are necessary conditions for the stability of the quasi-one-dimensional solutions. Because *μ* is negative (subcritical case), we focus on positive cubic cross-coupling parameter *c*_{r}, so that a quasi-linear instability can be induced during the interaction of both equations, as it has been shown in our previous paper [52]. We do not consider quintic cross-coupling terms for simplicity.

For the simulation of equations (2.1) and (2.2), we have used as numerical method explicit fourth order Runge–Kutta finite differencing with a rectangular grid of 700 points in *x* and 350 points in *y* along a grid spacing of d*x*=d*y*=0.1 (corresponding to a rectangular box size 70×35) and a time step d*t*=0.005. To avoid transients, we have waited until *T*∼500, corresponding to 10^{5} iterations.

## 3. Results and discussion

Here, we analyse the types of behaviour which result when the quasi-one-dimensional states have not yet reached their asymptotic shape as a function of time. The class of states considered here is depicted in figure 2*a*. It has to be contrasted with the asymptotic states used in reference [52] as initial conditions, which always lead to a unique outcome. In figure 1, we give an overview of the types of behaviour as a function of the approach velocity *v* and the destabilizing real part of the cross-coupling *c*_{r}. The range covered in the plot is 0.1<*v*<0.7 and 0.1<*c*_{r}<0.6. For *v* sufficiently large and *c*_{r} sufficiently small interpenetration is obtained with the final quasi-one-dimensional states reaching in size and shape their asymptotic behaviour. For *v* sufficiently small and *c*_{r} sufficiently large, we obtain non-unique results as the outcome of the collisions, whereas in the range in between these two limiting cases the result of collisions of two quasi-one-dimensional states is one compound state.

The possibility of a non-unique outcome as a result of collisions is intrinsically connected to the fact that the initial conditions used were not in the quasi-one-dimensional asymptotic state. That non-unique results might arise for non-asymptotic initial conditions has been briefly indicated in reference [52], where we used initial conditions for which the quasi-one-dimensional states were closer together (compare fig. 5*a* of [52]). The parts of figure 1 showing interpenetration and one compound state resemble closely those described in reference [52], but are slightly shifted to larger values of *c*_{r}.

To investigate the non-uniqueness in detail, we use series of snapshots, space time plots taken for a line along the crest of one of the initial quasi-one-dimensional states and three-dimensional plots of the asymptotic final states.

Figure 2 shows an example of the time evolution from an initial state of the class considered here (figure 2*a*) to one of the possible final states for these parameter values (*c*_{r}=0.6 and *v*=0.5). The start of the instability along the crest can be seen in figure 2*b* with the black dashed line in *y*-direction indicating the base line for the *y*–*t* plot shown as figure 3. Figure 2*c,d* shows the break-up of the quasi-one-dimensional state on the right into two stationary states localized in two dimensions and in figure 2*d,* we also see the development of the instability along the crest in the quasi-one-dimensional state on the left. Figure 2*e* shows the asymptotic result in time: four stationary localized solutions.

The time evolution for the initial break-up on the right is further elucidated in figure 3. The times for which the snapshots leading to figure 2*b*–*d* have been taken are indicated in figure 3. We note that the oscillatory behaviour observed for early times disappears in the asymptotic limit in time. A three-dimensional plot of the asymptotic result is presented in figure 4.

Starting with an initial condition of the same class, we can get several other types of asymptotic behaviour in time. We show two examples for the same parameter values for which figures 2–4 have been obtained, *c*_{r}=0.6 and *v*=0.5, in figure 5.

In the region of non-unique final results, there is also the possibility to obtain oscillatory states, a feature not encountered when starting with asymptotic stationary quasi-one-dimensional initial conditions. In figures 6 and 7, we give an example for *c*_{r}=0.5 and *v*=0.4. In figure 6*a,* we present a snapshot of the resulting asymptotic state showing a quasi-one-dimensional state and two oscillatory localized states. The white dashed line in figure 6 is used to generate the *y*–*t* plot given in figure 6*b*. Inspecting figure 6*b,* we see that this final result is characterized by one frequency for both localized compound states, but that the oscillations are not in phase. This feature is also brought out very clearly by the three-dimensional plot shown in figure 7, where the two oscillatory compound localized states can be seen to shed waves. We also note that the amplitude of the oscillatory compound localized states containing *A* and *B* is always smaller than the amplitude of the quasi-one-dimensional compound localized states. This feature is similar to the one seen for quasi-one-dimensional localized states and their compound states with localized states for one cubic–quintic complex Ginzburg–Landau equation for one amplitude *A* studied in reference [40]. We finally note that for the same parameter values we found that also one compound state composed of *A* and *B* can be left over as a result of the collision without any localized solution.

The results shown in this paper are asymptotic results, in particular figures 4 and 5 are asymptotic states. However, we emphasize that as a transient two circular dissipative solitons can merge upon collision into one circular dissipative soliton.

An interesting open question is whether spiral dissipative solitons (stable localized vortices) with topological charge/‘spin’ with *S*=1 and *S*=2 described in reference [53] can be generated by the type of collisions studied here. So far, we have only observed in our numerical investigations dissipative solitons with topological charge *S*=0.

For all the compound states containing *A* and *B*, it turns out that the maxima of |*A*| and |*B*| are located fairly close together in the asymptotic time limit. This is an observation we have already made for the case of collisions of quasi-one-dimensional states in their asymptotic shape as initial conditions [52]. We decided to exploit this observation by studying one cubic–quintic complex Ginzburg–Landau equation with effective parameters. Because we are investigating the case *c*_{i}=0, it turns out that the only effective parameter coming into play is an effective value for . For the behaviour shown in figures 6 and 7, this effective value for *β*_{r} assumes the value . In figure 8, we have plotted the resulting *y*–*t* plot. An intuitive picture for the concept of is given in §4.

## 4. *β* effective: an intuitive picture

In figure 1, we observe non-unique results for the cubic cross-coupling parameter *c*_{r} taking values 0.5 or 0.6. In addition, we do not observe localized solutions for *c*_{r}=0.4. The localized pulses obtained as part of these non-unique results are oscillating for *c*_{r}=0.5 and stationary for *c*_{r}=0.6. In this section, we present an intuitive explanation for the differences in behaviour.

All the solutions coming from the collisions of the counter-propagating cubic–quintic Ginzburg–Landau equations are composed by the two fields *A*(*x*,*y*,*t*) and *B*(*x*,*y*,*t*). Although the maxima of both fields are very close to each other, the distance between them never vanishes. We also note that the two fields *A* and *B* do not have the same phase. In a first approximation, we can neglect this distance between the maxima of |*A*| and |*B*| and also the phase difference, so that a non-moving pulse (either stationary or oscillating) should be a solution of the sum of equations (2.1) and (2.2):
4.1where . Numerical simulations of equation (4.1), starting with localized initial conditions, give as a result no stable pulses for , corresponding to *β*_{r}=1.0 and *c*_{r}=0.4, oscillating pulses for , corresponding to *β*_{r}=1.0 and *c*_{r}=0.5, and stationary pulses for , corresponding to *β*_{r}=1.0 and *c*_{r}=0.6.

Therefore, these results are qualitatively in agreement with what we have found and shown in figure 1. We note, however, that the frequency observed for (figure 8) is *f*=0.36, which is smaller than that observed for equations (2.1) and (2.2) and shown in figure 6 ( *f*=0.45). The difference in frequency can be traced back to the two approximations discussed before equation (4.1): the difference in distance between the maxima in *A* and *B* as well as the phase difference of the two fields *A* and *B*. Moreover, we observe that the frequencies of the oscillating pulses as part of the non-unique solutions (figure 1), depend on the approach velocity; the frequencies become smaller with increasing velocity.

## 5. Conclusion

In conclusion, we have analysed the interaction of quasi-one-dimensional dissipative solitons that are not in their asymptotic shape in time. We find non-unique outcomes as a result of the collisions for fixed parameters for sufficiently small approach velocities and sufficiently large interaction of the counter-propagating localized solutions. The non-uniqueness has been traced back to the occurrence of a modulation instability along the crest of the quasi-one-dimensional dissipative solitons. For quasi-one-dimensional dissipative solitons, which are not in their asymptotic shape before the collision, the quasi-linear instability does not occur simultaneously everywhere along the quasi-one-dimensional dissipative soliton. Among the solutions which arise owing to the non-uniqueness are one, two, three and four stationary localized solutions, oscillatory dissipative solitons as well as assemblies of quasi-one-dimensional dissipative solitons with stationary or oscillatory dissipative solitons.

## Authors' contributions

Both authors contributed equally to this paper by means of substantial contributions to conception and design, acquisition of data, analysis of results, drafting and final approval of the version to be published.

## Competing interests

We declare we have no competing interests.

## Funding

O.D. wishes to acknowledge the support of FONDECYT (project no. 1140139) and Universidad de los Andes through FAI initiatives. H.R.B. thanks the Deutsche Forschungsgemeinschaft for partial support of this work.

## Footnotes

One contribution of 13 to a theme issue ‘Topics on non-equilibrium statistical mechanics and nonlinear physics (II)’.

- Accepted June 10, 2015.

- © 2015 The Author(s)