Eur. Phys. J. B (2015) 88: 219 DOI: 10.1140/epjb/e2015-60537-y
THE EUROPEAN PHYSICAL JOURNAL B
Regular Article
Interaction of exploding dissipative solitons Orazio Descalzi1,2,a and Helmut R. Brand2,b 1 2
Complex Systems Group, Facultad de Ingenier´ıa y Ciencias Aplicadas, Universidad de los Andes, Av. Mons. ´ Alvaro del Portillo 12.455, Las Condes, Santiago, Chile Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany Received 1st July 2015 / Received in final form 17 July 2015 c EDP Sciences, Societ` Published online 7 September 2015 – a Italiana di Fisica, Springer-Verlag 2015 Abstract. We investigate the collisions of two counter-propagating exploding dissipative solitons (DSs). We demonstrate that six different outcomes can occur as a function of the nonlinear cross-coupling between the counter-propagating waves: complete interpenetration, one compound exploding DS as well as four types of two compound DSs that can be stationary, have one or two frequencies or are exploding. Since exploding DSs have been observed experimentally predominantly in nonlinear optics, we conjecture that our predictions for their interactions can be tested in laser systems.
1 Introduction A signature of conservative solitons is that they emerge unchanged from collisions except for a phase shift [1]. In non-conservative systems the existence of dissipative solitons (DSs) [2] is based on a balance of nonlinearity, dispersion, gain and loss. The field of dissipative solitons has been stimulated historically by the experimental observation of localized solutions near the onset of convection in binary fluid mixtures in rectangular [3–5] and annular [5,6] containers. Shortly thereafter the pioneering paper by Thual and Fauve [7] demonstrated numerically the stable existence of stationary localized solutions in the framework of the cubic-quintic complex Ginzburg-Landau equation. In the following the properties of stationary onedimensional DSs in the cubic-quintic complex GinzburgLandau equation were studied in more detail [8–11] including the existence of analytic solutions [12,13]. We note, however, that so far no stable analytic localized solution of the one-dimensional cubic-quintic complex GinzburgLandau equation has been reported. Experimentally it has been shown that partial annihilation, mutual annihilation, or a compound state of DSs are possible outcomes of collisions. In binary fluids, the collision between counter-propagating convective pulses has been studied as a function of the approach velocity. At higher velocities, partial annihilation has been observed, while for low velocities, the creation of bound states of pulses has been reported [14]. In a chemical system, the catalytic oxidation of CO on Pt (110), collisions of counter-propagating concentration waves exhibit mutual annihilation, partial annihilation, and interpena b
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etration [15,16]. Recently, in the context of biological physics, it has been reported that counter-propagating arc-shaped multicellular structures of non-chemotactic mutants present interpenetration [17]. Theoretically, collisions of dissipative solitons have been investigated by means of prototype equations, such as envelope and order parameter equations, or reactiondiffusion equations. In the frame of one-dimensional coupled cubic-quintic complex Ginzburg-Landau equations for counter-propagating waves, a dispersive-dissipative system, interpenetration and annihilation were for the first time reported in reference [18] (compare also Ref. [19] for a recent brief review). For the same equation and for annular geometry, collisions of pulses lead to many different outcomes, in particular to holes via front interaction [20]. Analyzing stochastic complex Ginzburg-Landau equations it was shown that partial annihilation is a noise-induced effect for collision of fixed-shaped pulses [21]. Interactions of two-dimensional solutions have been also investigated [22] (and references therein). Moreover, mutual annihilation and interpenetration of pulses have been found in excitable reaction-diffusion systems [23,24]. Recently the interpenetration of traveling bands in systems of deformable self-propelled particles has been studied [25]. Among the most fascinating DSs are exploding dissipative solitons. All parts of their spatio-temporal evolution are unstable but nevertheless they remain confined in space (time) keeping a certain frequency on average between explosions. They were first discovered in the cubic-quintic complex Ginzburg-Landau equation [26] and further characterized in references [27,28]. It was shown experimentally that a Kerr lens mode-locked Ti:sapphire laser can produce exploding solitons [29]. Further theoretical characterization leads to a transition from modulated to exploding dissipative solitons [30,31].
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In this paper, the collision of two counter-propagating exploding solitons, in two one-dimensional coupled cubicquintic complex Ginzburg-Landau equations, is studied. Unexpected results, such as the complete interpenetration of exploding solitons or the merging of two explosive compound states, are reported. Since very recently exploding solitons have been observed in a passively modelocked fiber laser [32] we suggest this system to test our predictions.
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∂t A − v∂x A = μA + (βr + iβi )|A|2 A + (γr + iγi )|A|4 A + (cr + ici )|B|2 A + (Dr + iDi )∂xx A, (1) ∂t B + v∂x B = μB + (βr + iβi )|B|2 B + (γr + iγi )|B|4 B + (cr + ici )|A|2 B + (Dr + iDi )∂xx B, (2) where A(x, t) and B(x, t) are complex fields and where we have discarded quintic cross-coupling terms for simplicity. A and B are slowly varying envelopes, βr is positive, and γr is negative in order to guarantee that the bifurcation is subcritical, but saturates to quintic order. We have carried out our numerical studies for the following values of the parameters, which we kept fixed for the present purposes: μ = −0.17, βr = 1, βi = 0.8, γr = −0.1, γi = −0.6, Dr = 0.125, Di = 0.5 and ci = 0. Positive values of Di correspond to the regime of anomalous linear dispersion. The time step dt used was typically dt = 0.005 and as a grid spacing we took dx = 0.08 and N = 1875 leading to a box size of L = 150. Time integration of equations (1) and (2) was performed using fourth order Runge-Kutta finite differencing.
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Fig. 1. Phase diagram in the plane strength of cubic crosscoupling of counter-propagating waves, cr versus approach velocity v. For the regime characterized by black triangles () we have interpenetration of exploding DSs containing one amplitude each. The state with two compound exploding DSs is denoted by a blue open square () and two compound states with one frequency are associated with blue solid squares (). Marked as red solid circles (•) are the stationary states of two compound DSs. As black solid triangles () we have depicted the compound state of exploding DSs as the result of the collision and for the state marked with a red open circle (◦) we obtain two compound states associated with two frequencies f1 and f2 . max( |A|,|B|) (a)
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3 Results and discussion In Figure 1 we present the phase diagram cr versus velocity v for the range −0.4 ≤ cr ≤ 0.4 and 0 ≤ v ≤ 0.5 in steps of 0.1. We find six different outcomes as a result of the collisions between two exploding solitons in the parameter regime studied: interpenetration, one compound state of exploding dissipative solitons containing A and B and four types of two compound states containing A and B simultaneously, namely stationary, a state with one frequency, a state with two frequencies and an explosive state. In Figure 2a we show the type of initial state of two exploding DSs used for all results discussed in the following. Figure 2b shows an x − t-plot of the interpenetration of two exploding DSs. We note that the interaction time is much longer than that for the interaction of colliding
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Fig. 2. The interpenetration of two exploding dissipative solitons. (a) Shows the initial state before the collision. Similar initial states are used for all runs. (b) The x − t-plot of an interpenetration is shown for cr = −0.1 and v = 0.3. We note the long interaction time until interpenetration occurs.
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Fig. 3. One compound exploding DS as an asymptotic result of collisions for cr = −0.2 and v = 0.3; (a) snapshot of the asymptotic state; (b) time series in the asymptotic time regime.
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stationary DSs [18,21]; in addition we observe that both exploding DSs need a fairly long time before they show explosions at their usual frequency again as it is observed in the asymptotic time regime. A frequent outcome of the collisions of two exploding DSs is one compound exploding DS as shown in Figure 3. As one can see from the phase diagram in Figure 1, this result is dominant for sufficiently negative (stabilizing) values of the real part of the cubic cross-coupling term cr . Figure 3a shows a snapshot of the asymptotic state while Figure 3b gives the time series in the asymptotic time regime. In Figure 4 we present four snapshots of the four different outcomes leading to two compound DSs in the asymptotic time regime. For large enough values of the cubic cross-coupling term cr always a stationary two compound state is obtained. As one can see from inspection of the the phase diagram given in Figure 1, two compound DSs with two frequencies and two compound exploding DSs occupy only a fairly small region in the phase diagram. In Figures 5 and 6 x − t-plots for the four different types of two compound states in the asymptotic time regime are shown. We note that for the state with one fairly large frequency a different time resolution is needed to visualize this frequency in the x − t-plot. This type of behavior is reminescent of the time resolution required for the explosive dissipative DS for the case of one field A [30]. The mechanism of generation of the four types of two compound DSs observed is of the same nature as the one described recently [33] for the collisions of stationary DSs in one and two spatial dimensions. Making use of the quasilinear growth rate A˙ = σA with σ = (μ + cr |B|2 ), we conclude that for sufficiently large destabilizing values of cr a positive growth rate can arise for the regions where |B| is nonvanishing and vice versa for |A| and |B| interchanged. Therefore in the final two compound DSs |A| and |B| are contained with equal amplitude in each part
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Fig. 4. The four different outcomes of collisions of exploding DSs leading to two compound DSs are shown as snapshots in the asymptotic time regime: (a) stationary case, cr = 0.4, v = 0.5: (b) two compound exploding DSs, cr = 0.2, v = 0.5; (c) two compound DSs with one frequency, cr = 0.2, v = 0.4; (d) two compound DSs with two frequencies, cr = 0.1, v = 0.2.
of the compound states. Thus the mechanism of quasilinear growth is quite general and can lead to the four types of two compound DSs described here. As already observed for the exploding DSs for one field A this type of dissipative solitons shows a meandering motion in the case of one spatial dimension [30,34]. In two spatial dimensions a diffusion-like behavior was found
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for asymmetric exploding 2D DSs, while the azimuthally symmetric DSs showed no motion of their center of mass even for long times [31,35]. In the 2D case this different behavior of azimuthally symmetric exploding DSs versus asymmetric exploding DSs had as a consequence that azimuthally symmetric exploding 2D DSs could form bound states with quasi-one dimensional DSs. In one spatial dimension we have observed a different consequence of the meandering motion for cr = 0.2 and v = 0.5, which is shown in Figure 7. For the explosive two compound DS state it is possible that a coalescence occurs due to the meandering motion as plotted in Figure 7. Thus one obtains for fixed parameters a non-unique outcome in the asymptotic time regime, namely the one displayed in Figures 4b and 5b and also the one presented in Figure 3. Inspecting Figure 7 one sees that as a precur-
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Fig. 7. x − t-plot for the coalescence of two exploding compound DSs into one exploding compound DS for cr = 0.2, v = 0.5, and T = 100. Note the complex interaction preceding the collapse to one exploding compound DS.
sor to coalescence first one of the explosive compound DSs splits into two peaks (compare the loop type behavior in the x − t-plot in Fig. 7). The three resulting peaks then undergo a giant explosion for an extended period of time before settling down to one compound exploding DS. The splitting into three peaks has always been observed as a precursor to coalescence. We also would like to add a note on the time scales involved. Typically in obtaining the results displayed in Figure 1 we waited for time T = 5000 or 106 iterations. All other outcomes of collisions were always reproducible and only the case of exploding two compound states was different in the sense of non-uniqueness for the same class of initial conditions. We trace this fact back to the meandering motion of exploding dissipative solitons.
4 Conclusion In summary we have demonstrated that the collisions of two counterpropagating exploding dissipative solitons (DSs) can lead to six different outcomes for the range of approach velocities and the strength of the cubic crosscoupling between the counter-propagating waves studied. The resulting scenarios comprise the complete interpenetration of propagating exploding DSs as well as five types of compound states. As an intriguing result we find that in some cases a two compound exploding DS can transform into a one compound exploding DS after a rather long transient due to the meandering motion of exploding DSs thus leading to a non-unique outcome of collisions of exploding dissipative solitons. To test the predictions made experimentally, it appears natural to exploit the know-how already accumulated during the study of exploding DSs
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in nonlinear optics, namely for a mode-locked Ti:sapphire laser [29] and very recently for a passively mode-locked fiber laser [32].
Author contribution statement Both authors contributed equally to this paper. 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.
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