Evidence of Diffusing Dissipative Solitons - APS Link Manager

week ending 26 OCTOBER 2012

PHYSICAL REVIEW LETTERS

PRL 109, 178303 (2012)

Model of a Two-Dimensional Extended Chaotic System: Evidence of Diffusing Dissipative Solitons Carlos Cartes,1 Jaime Cisternas,1 Orazio Descalzi,1,2,* and Helmut R. Brand2 1

Complex Systems Group, Facultad de Ingenierı´a y Ciencias Aplicadas, Universidad de los Andes, Avenida San Carlos de Apoquindo 2200, Santiago, Chile 2 Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany (Received 9 July 2012; published 23 October 2012)

We investigate a two-dimensional extended system showing chaotic and localized structures. We demonstrate the robust and stable existence of two types of exploding dissipative solitons. We show that the center of mass of asymmetric dissipative solitons undergoes a random walk despite the deterministic character of the underlying model. Since dissipative solitons are stable in two-dimensional systems we conjecture that our predictions can be tested in systems as diverse as nonlinear optics, parametric excitation of granular media and clay suspensions, and sheared electroconvection. DOI: 10.1103/PhysRevLett.109.178303

PACS numbers: 82.40.Bj, 05.70.Ln, 42.65.Sf, 47.20.Ky

Stable localized structures, a hallmark of dissipative nonequilibrium systems, have been found [1–22], both experimentally [3–5,11,14,16,18–20], and theoretically [1,2,6–10,12,13,15,17,21,22], in systems as diverse as surface reactions [5], binary fluid convection [3,4], sheared electroconvection in liquid crystals [20], and nonlinear optics [14,18,19]. Such stable localized solutions, also called dissipative solitons, do not only stably exist in one, but also in two spatial dimensions [1,5,6,8–11,13–16,20,21], setting them apart from systems without dissipation: solitons arising in nondissipative onedimensional systems are, in most cases, well known [23] to be unstable to perturbations in a second direction. Exploding dissipative solitons found by Akhmediev et al. for anomalous linear dispersion in the cubic-quintic complex Ginzburg-Landau equation [17], have been further characterized experimentally [18] and theoretically [24–27] in one spatial dimension including most recently the influence of a small amount of noise, which was shown to lead to qualitative changes [28]. In addition, it has been shown that a number of temporally regular localized solutions emerge as the bifurcation parameter , the distance from linear onset, is increased towards the regime for which explosive dissipative solitons are found [26,27]. In two-dimensional systems, Soto-Crespo et al. found first that pulsating beams can be transformed into exploding dissipative solitons [29]. In this Letter, the spatiotemporal dynamics of explosive dissipative solitons in the two-dimensional cubic-quintic complex Ginzburg-Landau equation with positive linear dispersion is studied as a function of the bifurcation parameter and the cubic nonlinear refractive index i . The spatial symmetry or asymmetry is described by an analog of the inertia tensor and the analysis of the temporal evolution is based on more than 30 000 explosions. To characterize quantitatively the spatiotemporal transition from azimuthally symmetric to asymmetric explosions we introduce 0031-9007=12=109(17)=178303(5)

a suitable order parameter. We emphasize that this transition has no analog in one-dimensional systems. As a remarkable fact we find that the center of mass of asymmetric dissipative solitons undergoes a random walk despite the deterministic nature of the underlying model. The two-dimensional cubic-quintic complex GinzburgLandau equation reads @t A ¼ A þ ðr þ ii ÞjAj2 A þ ðr þ ii ÞjAj4 A þ ðDr þ iDi Þr2 A;

(1)

where Aðx; y; tÞ is a complex field, r is positive, and r is negative in order to guarantee that the bifurcation is subcritical, but saturates to quintic order. The diffusion coefficient Dr and the linear dispersion Di are positive. In nonlinear optics, where stable dissipative propagating spatial solitons are the result of a balance between gain and loss of the media as well as the balance between diffraction and nonlinearities, the cubic-quintic complex GinzburgLandau equation reads D 2 r c þ j c j2 c þ j c j4 c 2 ? ¼ i c þ i"j c j2 c þ ir2? c þ ij c j4 c :

ic z þ

(2)

Here c ðx; y; zÞ is the normalized envelope of the electrical field, r? 2 ¼ @2x þ @2y is the two-dimensional transverse Laplacian, z is the propagation distance, D is the diffraction coefficient, stands for linear losses, " and are the cubic and quintic nonlinear gain and absorption, respectively, is the saturation coefficient of the Kerr nonlinearity, and stands for spectral filtering in the medium. Comparing Eqs. (1) and (2) we identify t ¼ z, ¼ , r ¼ ", i ¼ 1, r ¼ , i ¼ , Dr ¼ , Di ¼ D=2. In this Letter and in most cases simulations have been carried out using a square box of size 100 100, L 512 512 points, dx ¼ dy ¼ 512 with L ¼ 100, and dt ¼ 0:0025. For the time integration we implemented a

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Ó 2012 American Physical Society



PRL 109, 178303 (2012)

pseudospectral split-step scheme using a fourth order Runge-Kutta method to integrate the nonlinear part of Eq. (1). To accelerate our codes we used the PYCUDA extension [30]. Equation (1) has seven parameters. However, rescaling the modulus of the complex field A, jAj, time, and space, three of these parameters can be fixed without loss of generality. In our case these parameters are: r ¼ 1, r ¼ 0:1, and Dr ¼ 0:125. For the remaining four free parameters we proceed as follows: we fixed i ¼ 0:6, and Di ¼ 0:5, while , the distance from linear onset, and i , the cubic nonlinear refractive index, remained free as shown in Fig. 1. As initial conditions we used localized solutions occupying around 10% of the area of the box. Periodic or no-flux boundary conditions lead to the same results asymptotically in time. Figure 1 presents the overview of our qualitative main results we find for exploding dissipative solitons in two spatial dimensions. The phase diagram presents the stable localized solutions observed for a large range of the bifurcation parameter, , which is varied from ¼ 0:55 to ¼ 0:30. On the ordinate we have plotted the cubic part of the nonlinear refractive index, which is the second parameter we have varied. In this phase diagram we observe a total of five transitions, three of which have not been observed in one dimension. This pertains to all transitions involving azimuthally symmetric exploding

dissipative solitons, which have no analog in one spatial dimension. The most interesting transition is clearly the one from azimuthally symmetric to asymmetric exploding solitons. This transition as well as the two different types of stable localized solutions involved will be studied in detail in the following. The other types of localized solutions found are stable stationary pulses as they have been known to exist stably for the cubic-quintic Ginzburg-Landau equation in one and two spatial dimensions [1,6]. We also note that Fig. 1 contains a direct transition between collapse (all classes of bounded initial conditions including localized ones decay as a function of time) and azimuthally symmetric explosions. In Fig. 2 we present four snapshots showing important characteristic features of the time evolution of azimuthally symmetric exploding dissipative solitons. While there is a fairly narrow distribution of times between consecutive explosions, this process is not associated with a fixed frequency. Figure 2 has been obtained in the asymptotic time regime after all initial transients have disappeared. It is important to emphasize that for the azimuthally symmetric exploding solitons, which have no analog in one spatial dimension, the azimuthal symmetry is preserved not only after many explosions, but also when the amplitude is close to its maximum during the explosions [see Figs. 2(b) and 2(c)].

β 0.9

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0.85

0.8

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FIG. 1 (color online). Phase diagram showing the observed stable patterns as a function of the cubic nonlinear refractive index, i , on the ordinate versus the bifurcation parameter . We show the parameter range from 0:55 to 0:3. It includes: (a) solutions with amplitude zero (solid triangles), (b) stationary localized solutions (solid squares), (c) azimuthally symmetric exploding localized solutions shown as black solid circles, and asymmetric exploding localized solutions given by open circles. All data points shown have been obtained with numerical runs with a duration T ¼ 6 103 corresponding to 6 105 iterations. We note that the boundaries between the different types of behavior shown are not completely sharp because of a small hysteresis. In Fig. 7 we show this phenomenon for the transition from azimuthally symmetric to asymmetric explosions for i ¼ 0:8.

FIG. 2 (color online). Four snapshots are plotted characteristic of the time evolution of an azimuthally symmetric exploding dissipative soliton. The parameters chosen for these snapshots are ¼ 0:5 and i ¼ 1:0. All other parameter values are the same as given in the main text. Snapshots (a) and (d) show the exploding dissipative soliton before and after an explosion while in (b) the explosive behavior is starting and in (c) it is close to its maximum. Note that the azimuthal symmetry is preserved during the explosion.

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(a)

(b)

(c)

(d)

FIG. 3 (color online). Four snapshots are plotted characteristic of the time evolution of an asymmetric exploding dissipative soliton. The parameters chosen for these snapshots are ¼ 0:4 and i ¼ 1:0. All other parameter values are the same as given in the main text. Snapshots (a) and (d) show the exploding dissipative soliton before and after an explosion while in (b) the explosive behavior is starting and in (c) it is close to its maximum. We note the strong asymmetry arising during the explosion as it is particularly evident in the snapshots shown in (b) and (c).

the shape of the asymmetric explosions as well as the orientation of the entire object vary even in the long time limit. The center of mass of the distribution ¼ jAðx; y; tÞj undergoes a random walk as it is shown in Fig. 4. To analyze the time evolution of both types of exploding dissipative solitons quantitatively we show in Fig. 5 x-t plots of a cross section, with azimuthally symmetric explosions on the left and asymmetric explosions on the right. The times corresponding to the snapshots presented in Figs. 2 and 3 have been indicated in both cases. One reads off right away that asymmetric explosions are more frequent when all parameters, except for , the distance from linear onset are varied. This is also intuitively clear, since the systems has access to more degrees of freedom as the linear onset is approached. An increase in the number of explosions as is increased towards the linear onset has also been found in 1 D [26] reflecting the fact that the zero solution becomes more sensitive to perturbations. The asymmetry of the explosions is clearly brought out also in the x-t plots. Compare, for example, the symmetry or

60

(b) (c)

0 10 x

y

-20 -40 -20

0 x

20

40

FIG. 4 (color online). Random walk of an asymmetric exploding soliton, for ¼ 0:31 and i ¼ 0:8. Red (grey) color stands for the first 4 105 time units and black color for the last 4 105 time units. The initial condition is taken at the center of the box.

(a)

(b)

t

0

(d) (c)

(d)

40 20

B

A

In Fig. 3 we show four snapshots of the time evolution of asymmetric exploding dissipative solitons. In particular for large amplitudes as they prevail in Figs. 3(b) and 3(c), the asymmetry is strongly pronounced and one can clearly see the qualitative differences with the case of azimuthally symmetric explosions shown in Fig. 2. We note that in the long time limit this asymmetry is not reduced and that

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PRL 109, 178303 (2012)

(a)

t

40 10 x

40

FIG. 5 (color online). x-t plots of a cross section are shown in the asymptotic time regime for T ¼ 60 for azimuthally symmetric (A, on the left) as well as for asymmetric (B, on the right) exploding dissipative solitons. The parameters are the same as for Figs. 2 and 3, respectively: A ¼ 0:5, i ¼ 1:0, B ¼ 0:4, i ¼ 1:0. The labels (a), (b), (c), and (d) in A and B correspond to the times of the snapshots plotted in Figs. 2 and 3: For the time series of the azimuthally symmetric exploding dissipative soliton shown on the left (A) they read: (a) T ¼ 4:21, (b) T ¼ 21:82, (c) T ¼ 22:56, and (d) T ¼ 35:21; for the time series of the asymmetric exploding dissipative soliton shown on the right (B) they are: (a) T ¼ 17:98, (b) T ¼ 34:71, (c) T ¼ 36:20, and (d) T ¼ 38:68.

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PRL 109, 178303 (2012)


0.7

g(t)

0.1

(a)

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0.08

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t 0

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15000

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I(τ)

-0.38 -0.36 -0.34 -0.32

30000

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µ

FIG. 7 (color online). The order parameter characterizing the transition from azimuthally symmetric to asymmetric explosions, hIðÞi, is plotted as a function of for i ¼ 0:8. Some hysteresis is present at the onset of the transition.

(b)

0.08 0.06 0.04

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1300

2300

3300

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6300

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FIG. 6 (color online). (a) A temporal measure of the asymmetry of the explosions is given by gðtÞ ¼ ðþ Þ=þ . Asymmetric explosions appear in groups separated by symmetric or weakly asymmetric explosions. The plot is for ¼ 0:31 and i ¼ 0:8. (b) Plot of IðÞ for a group shown in (a) (grey color). Since IðÞ remains almost constant inside the group, the temporal average hIðÞi ¼ 0:08 is considered as the order parameter.

asymmetry in the time evolution near the times marked as (b) and (c) in both cases. To analyze quantitatively the transition from azimuthally symmetric to asymmetric explosions we calculate the analog of the inertia tensor for the exploding R 2 dissipative ¼ y dxdy, Ixy ¼ soliton at each time defined by I xx R R xydxdy, Iyy ¼ x2 dxdy, with x, y measured with respect to the center of mass coordinates. At each time, using principal axes, the inertia tensor reduces to Ix0 x0 ¼ þ , Iy0 y0 ¼ , and Ix0 y0 ¼ 0, where ¼ 12 ðIxx þ 2 1=2 Þ. Symmetric objects require Iyy ½ðIxx Iyy Þ2 þ 4Ixy þ ¼ 0, otherwise they will be considered as asymmetric. Then we define gðtÞ ðþ Þ=þ as a temporal measure of the asymmetry of the exploding dissipative solitons. Figure 6(a) shows that asymmetric explosions appear in groups separated by symmetric or weakly asymmetric explosions. This figure contains around 600 explosions, where roughly one third corresponds to asymmetric explosions. The temporal distance between groups is irregular, typical for intermittency. By increasing the bifurcation parameter the distance between groups tends to decrease. To characterize the transition from azimuthally symmetric to asymmetric explosions, we introduce the R quantity IðÞ, defined inside a group: IðÞ ¼ 0 gðtÞdt=. Figure 6(b) shows IðÞ inside a group [depicted by grey color in Fig. 6(a)], where is the variable characterizing the time inside the group. We note that IðÞ remains almost

constant inside the groups. As shown in Fig. 7 the temporal average hIðÞi emerges as a suitable order parameter describing the transition from symmetric to asymmetric explosions. We emphasize that this picture is based on the spatiotemporal analysis of more than 30 000 explosions. In summary we have demonstrated the robust and stable existence of two types of exploding dissipative solitons in two spatial dimensions for a pattern forming system with dissipation and dispersion. We have introduced an order parameter to characterize the transition between the two types of dissipative solitons and we have shown that asymmetric dissipative solitons undergo a random walk for a deterministic model. The predictions made must now be tested experimentally in pattern formation. Suitable systems come from fields as diverse as nonlinear optics [14,18], parametric excitation of granular media and clay suspensions [11,16], and sheared electroconvection [20]. O. D., J. C., and C. C. wish to thank the support of FONDECYT (Projects No. 1110360 and No. 3110028) and Universidad de los Andes through FAI initiatives. H. R. B. thanks the Deutsche Forschungsgemeinschaft for partial support.

*[email protected] [1] O. Thual and S. Fauve, J. Phys. (Les Ulis, Fr.) 49, 1829 (1988). [2] H. R. Brand and R. J. Deissler, Phys. Rev. Lett. 63, 2801 (1989). [3] P. Kolodner, Phys. Rev. A 44, 6448 (1991). [4] P. Kolodner, Phys. Rev. A 44, 6466 (1991). [5] H. H. Rotermund, S. Jakubith, A. von Oertzen, and G. Ertl, Phys. Rev. Lett. 66, 3083 (1991). [6] R. J. Deissler and H. R. Brand, Phys. Rev. A 44, R3411 (1991). [7] R. J. Deissler and H. R. Brand, Phys. Rev. Lett. 72, 478 (1994). [8] H. R. Brand and R. J. Deissler, Physica (Amsterdam) 204A, 87 (1994).

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[9] R. J. Deissler and H. R. Brand, Phys. Rev. E 51, R852 (1995); Phys. Rev. Lett. 74, 4847 (1995). [10] H. R. Brand and R. J. Deissler, Physica (Amsterdam) 216A, 288 (1995). [11] P. B. Umbanhowar, F. Melo, and H. L. Swinney, Nature (London) 382, 793 (1996). [12] V. V. Afanasjev, N. Akhmediev, and J. M. Soto-Crespo, Phys. Rev. E 53, 1931 (1996). [13] T. Ohta, Y. Hayase, and R. Kobayashi, Phys. Rev. E 54, 6074 (1996). [14] V. B. Taranenko, K. Staliunas, and C. O. Weiss, Phys. Rev. A 56, 1582 (1997). [15] H. Sakaguchi and H. R. Brand, Physica (Amsterdam) 117D, 95 (1998). [16] O. Lioubashevski, Y. Hamiel, A. Agnon, Z. Reches, and J. Fineberg, Phys. Rev. Lett. 83, 3190 (1999). [17] J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, Phys. Rev. Lett. 85, 2937 (2000). [18] S. T. Cundiff, J. M. Soto-Crespo, and N. Akhmediev, Phys. Rev. Lett. 88, 073903 (2002). [19] E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, Phys. Rev. Lett. 90, 253903 (2003).


[20] P. Tsai, S. W. Morris, and Z. A. Daya, Europhys. Lett. 84, 14003 (2008). [21] A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo, Phys. Rev. A 77, 033840 (2008). [22] O. Descalzi, J. Cisternas, D. Escaff, and H. R. Brand, Phys. Rev. Lett. 102, 188302 (2009). [23] A. C. Newell, Solitons in Mathematics and Physics (SIAM, Philadelphia, 1985). [24] N. Akhmediev, J. M. Soto-Crespo, and G. Town, Phys. Rev. E 63, 056602 (2001). [25] N. Akhmediev and J. M. Soto-Crespo, Phys. Lett. A 317, 287 (2003); N. Akhmediev and J. M. Soto-Crespo, Phys. Rev. E 70, 036613 (2004). [26] O. Descalzi and H. R. Brand, Phys. Rev. E 82, 026203 (2010). [27] O. Descalzi, C. Cartes, J. Cisternas, and H. R. Brand, Phys. Rev. E 83, 056214 (2011). [28] C. Cartes, O. Descalzi, and H. R. Brand, Phys. Rev. E 85, 015205 (2012). [29] J. M. Soto-Crespo, N. Akhmediev, N. Devine, and C. Mejı´a-Corte´s, Opt. Express 16, 15 388 (2008). [30] A. Kloeckner, N. Pinto, Y. Lee, B. Catanzaro, P. Ivanov, and A. Fasih, Parallel Comput. 38, 157 (2012).

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