Exploding dissipative solitons in the cubic-quintic ... - Springer Link

Eur. Phys. J. Special Topics 223, 2145–2159 (2014) © EDP Sciences, Springer-Verlag 2014 DOI: 10.1140/epjst/e2014-02255-2

THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS

Review

Exploding dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in one and two spatial dimensions A review and a perspective C. Cartes1 , O. Descalzi1,2,a , and H.R. Brand2 1 2

Complex Systems Group, Facultad de Ingenier´ıa y Ciencias Aplicadas, Universidad de los Andes, Av. Monse˜ nor Alvaro del Portillo 12455, Las Condes, Santiago, Chile Department of Physics, University of Bayreuth, Bayreuth, Germany Received 11 April 2014 / Received in final form 18 August 2014 Published online 24 October 2014 Abstract. We review the work on exploding dissipative solitons in one and two spatial dimensions. Features covered include: the transition from modulated to exploding dissipative solitons, the analogue of the Ruelle-Takens scenario for dissipative solitons, inducing exploding dissipative solitons by noise, two classes of exploding dissipative solitons in two spatial dimensions, diffusing asymmetric exploding dissipative solitons as a model for a two-dimensional extended chaotic system. As a perspective we outline the interaction of exploding dissipative solitons with quasi one-dimensional dissipative solitons, breathing quasi onedimensional solutions and their possible connection with experimental results on convection, and the occurence of exploding dissipative solitons in reaction-diffusion systems. It is a great pleasure to dedicate this work to our long-time friend Hans (Prof. Dr. Hans J¨ urgen Herrmann) on the occasion of his 60th birthday.

1 Introduction Stable spatially localized solutions in one and two dimensions have emerged as a hallmark of dissipative and driven nonequilibrium systems (compare Refs. [1, 2] for a recent exposition of the subject). They have been observed in a number of experimental systems [3–14] as diverse as binary fluid convection, surface reactions, nonlinear optics and starch suspensions. In addition, their stable existence and their properties have been investigated theoretically for a number of prototype and model equations [15–35]. They are frequently called dissipative solitons (DS) [36] to emphasize that they are characteristic of driven dissipative nonequilibrium systems and are much more robust and prevalent than ordinary solitons arising for nondissipative systems [37], which are typically unstable to transverse perturbations in a second or third direction. The term DS is reserved for localized solutions in space for nonequilibrium a

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systems with a thermodynamic driving force and dissipation that live indefinitely. By spatially localized we mean exponential decay in space. Quite recently it has also been pointed out that DS can serve as fundamental building blocks for the world of living organisms, since they can sustain localized flows of energy and matter [38]. Here we review the properties of a specific class of of DS, namely of exploding dissipative solitons, which have been discovered by Soto-Crespo et al. [27] for anomalous linear dispersion in the cubic-quintic complex Ginzburg-Landau equation in nonlinear optics. Shortly thereafter they were found also experimentally by Cundiff et al. [11]. Over the last ten years or so many of the properties of these localized objects, for which all parts of their time evolution are unstable, but which nevertheless stay localized and reappear with a certain frequency on average, have been studied in detail, both for the optical version of the cubic-quintic complex Ginzburg-Landau equation [39–42] as well as for the version of the cubic-quintic complex Ginzburg-Landau equation [43–48] as it arises as an envelope equation near the onset of a weakly inverted bifurcation to traveling waves [49, 50]. We just mention here two of the recent developments. While it has been shown for counter-propagating stationary DS that noise plays a key role in the outcome of the interaction (interpenetration, annihilation and partial annihilation) [35] as well as for the interpretation of experimental results [5–7], it turns out – as we have demonstrated recently [45] – that a small amount of noise can trigger the transition from an oscillatory DS to an exploding DS via various routes. Details will be given in the next section. Another development concerns the incorporation of nonlinear gradient terms into the cubic-quintic complex Ginzburg-Landau equation [49, 50]. While the physical consequences of such terms have been investigated a number of years ago for stationary DS [51] and for breathing DS [52], their effect has been studied recently also for exploding dissipative solitons in the nonlinear optics version of the cubic-quintic complex Ginzburg-Landau equation (including higher order derivatives as well) [53–55]. We note that the spatially localized exploding DSs must be distinguished from the various types of space-filling solutions reviewed in ref. [56] predominantly for the cubic CGL equation. We also emphasize that the exploding dissipative solitons reviewed here are of a completely different nature when compared to the fronts and the unstable fixed shape (in the modulus |A|) solutions studied in Ref. [57]. Even today the stable fixed shape solutions pioneered in Ref. [15] are only known numerically and not analytically exactly. In summary, exploding dissipative solitons (DSs) form a very interesting and challenging subclass of DSs. In the field of nonlinear optics the cubic-quintic complex Ginzburg-Landau equation takes the form [36, 39] D ψtt + |ψ|2 ψ + ν|ψ|4 ψ 2 = iδψ + iε|ψ|2 ψ + iβψtt + iμ|ψ|4 ψ,

iψz +

(1)

with the complex field ψ(z, t). In the following we will use predominantly the notation used for the description of instabilities. In this case the complex subcritical cubic-quintic Ginzburg-Landau equation in two spatial dimensions reads: ∂t A = μA + (βr + iβi )|A|2 A + (γr + iγi )|A|4 A +(Dr + iDi )(∂xx + ∂yy )A,

(2)

where A(x, y, t) is a complex field. In writing down this equation we have already transformed into the moving frame. To guarantee saturation to quintic order, γr is

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Table 1. Conversion of the coefficients of the cubic-quintic Ginzburg-Landau equation used in nonlinear optics versus the version used for the description of instabilities. Nonlinear optics δ ε 1 μ ν β D/2

Instability notation μ βr βi γr γi Dr Di

taken to be negative, while βr > 0 to have a weakly inverted bifurcation. The dynamical system corresponding to Eq. (2) has only one fixed point, A ≡ 0. Depending on μ one can have, in addition, one stable and one unstable limit cycle. The diffusion coefficient, Dr is assumed to be positive. In the limit Dr → 0 the localization is suppressed and one obtains a spatially homogeneous solution. In the spirit of an envelope equation, the fast spatial and temporal variations have already been split off when writing down Eq. (2). To compare with measurable quantities such as, for example, concentration and velocity variations in fluid dynamics, these rapid variations must be taken into account [49, 58, 59]. When making the replacements z → t and t → x in Eq. (1) an equation of the structure of Eq. (2) arises. To make the comparison between the two notations used in Eqs. (1) and (2) as easy as possible, we list the necessary identifications in Table 1.

2 Overview of main results 2.1 Transitions from stationary DS to explosions in one spatial dimension We investigate the transition to exploding dissipative solitons in Eq. (2) for one spatial dimension [43]. We keep all parameters fixed except for the bifurcation parameter μ (the distance from linear onset). The parameter values are βr = 1, βi = 0.8, γr = −0.1, γi = −0.6, Dr = 0.125, and Di = 0.5. Our initial conditions (IC) are localized but only rather precisely symmetric IC evolve in the symmetric branch, otherwise solutions belong to the asymmetric branch (compare Fig. 1). As a numerical method we use an explicit fourth-order Runge-Kutta finite differencing with a box size L = N dx = 50 along a grid spacing of dx = 0.08 and a time step dt = 0.005 to guarantee numerical stability. For μ < −1.23 any IC decreases until A = 0. For −1.23 < μ < −0.228 any IC, either symmetric or asymmetric, in phase or not in phase, evolves to a perfect (in modulus) symmetric stationary pulse. As μ is increased, in the range −0.226 < μ < −0.202, a rather precisely symmetric IC evolves to a symmetric (and in phase) (compare Fig. 2a) oscillating solution characterized by one frequency f1 , as it can be observed in the x − t plot shown in Fig. 3a. When μ is increased above μ = μc ∼ −0.202 a slow modulation emerges giving rise to an oscillating pulse characterized by two frequencies (f1 , f2 ), where f1 f2 . A typical x − t plot of the modulus |A| is shown in Fig. 3b. The symmetric branch ends at μ ∼ −0.183. Above this value of μ any IC evolves to an exploding dissipative soliton whose parts are unstable but nevertheless it remains confined in space. Its

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Fig. 1. Phase diagram showing the hysteretic transition to exploding dissipative solitons. Collapse: any IC leads to |A| = 0, ST: stationary pulses, f1 : oscillating pulses with one frequency, f1 , f2 : oscillating pulses with two frequencies. The parameter values are βr = 1, βi = 0.8, γr = −0.1, γi = −0.6, Dr = 0.125, Di = 0.5, dx = 0.08, and dt = 0.005. At μ = μc a continuous transition from the one frequency state f1 to the two frequencies state (f1 , f2 ) takes place.

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characteristic x − t plot is shown in Fig. 3d. As the value of μ is decreased below μ ∼ −0.183 we stay in the asymmetric branch and we obtain oscillatory pulses (not in phase) characterized by two frequencies (a typical x−t plot is shown in Fig. 3c). By further decreasing of μ we get a pulse (not in phase) characterized by one frequency (compare Fig. 2b). For values of μ ∼ −0.228 the system jumps to stationary pulses, which belong to the symmetric branch. Thus the transition to exploding dissipative solitons arises via a hysteretic transition, as it is shown in Fig. 1. The dynamics of explosions is characterized as follows: a perturbation grows in one wing of the oscillating pulse (shown in Fig. 4a). Once the perturbation overcomes a certain threshold (the system is subcritical) a new peak arises and interacts with the previous main peak (shown in Fig. 4b). The compound object reaches a maximum width (shown in Fig. 4c) and then it collapses to one pulse. Afterwards, a perturbation starts growing in the opposite wing (shown in Fig. 4d). The time between explosions is not fixed but it obeys a narrow temporal distribution.

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Fig. 4. Dynamics of one explosion. (a) A perturbation is growing in the left wing and after overcoming a threshold a new peak is created. (b) The new peak interacts with the main peak. (c) The compound object reaches a maximum width. (d) After a quick collapse to one pulse a new perturbation is growing in the right wing. μ = −0.17. The other parameter values are as for Fig. 1.

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Fig. 5. (a) Time series for the in-phase state f1 with one frequency ω1 = 16.17 (and its harmonics). ε = −2 · 10−4 . (b) Power spectrum of times series shown in (a). (c) Time series for the in-phase state with two frequencies (f1 , f2 ) with frequencies ω1 = 16.17, ω2 = 0.182, (and its harmonics). ε = 2 · 10−4 . Note that the temporal scale is six times those shown in figure (a). (d) Power spectrum of times series shown in (c). Note the different scales in frequency between figures (b) and (d). (e) Time series for an exploding dissipative soliton. ε = 2.5 · 10−2 . (f) For the onset of the appearance of exploding dissipative solitons the power spectrum shows signatures of low-dimensional chaos. Time series correspond to |A(x = 30, t)| (compare Fig. 2). ε = μ − μc .

2.2 Ruelle-Takens route for spatially localized solutions In this section we study the analog of the Ruelle-Takens route for spatially localized solutions [44]. As a numerical method we consider a pseudospectral split-step scheme, with a box size L = 50, a grid spacing dx = 0.08 and a time step dt = 0.005. For this numerical method μc = −0.2316 (compare Fig. 1). At μ = μc a continuous transition from one frequency pulses to two frequencies pulses takes place. The dynamics of one or two frequencies, or explosions, occur in a high-dimensional system. Nevertheless, we can project this dynamics onto a lower dimensional space computing time series for the in-phase state f1 with one frequency ω1 (shown in Fig. 5a), for the in-phase state


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with two frequencies (f1 , f2 ) (shown in Fig. 5c), and for exploding dissipative solitons (shown in Fig. 5e). Along with time series we have computed their corresponding power spectra. Figures 5b and 5d show the power spectra for the states with one and two frequencies, respectively. Figure 5f shows the power spectrum at the onset of the appearance of exploding dissiaptive solitons. There one can note the features characteristic of low-dimensional chaos including its characteristic low-frequency behavior. We have shown the existence of the following route for spatially localized solutions: one frequency, two frequencies, and then chaotic exploding dissipative solitons. We did not observe a third frequency before the appearance of chaos. This suggests an analog of the Ruelle-Takens-Newhouse route to chaos for localized states [60–62]. 2.3 Exploding dissipative solitons induced by noise Noise triggers many interesting and surprising effects in topics dealing with localized structures. For instance, recently it has been pointed out that noise induces partial annihilation of two counter-propagating dissipative pulses [35]. Partial annihilation has been observed experimentally in the context of hydrodynamics and chemical reactions over surfaces [5, 6, 63]. In this section we investigate the influence of noise on stationary pulses, oscillating pulses with one frequency, oscillating pulses with two frequencies, and exploding DSs. Thus, we study a stochastic cubic-quintic complex Ginzburg-Landau equation ∂t A = μA + (βr + iβi )|A|2 A + (γr + iγi )|A|4 A + (Dr + iDi )∂xx A + η ξ.

(3)

The stochastic forcing ξ(x, t) denotes white noise with the properties ξ = 0, correlations ξ(x, t) ξ(x , t ) = 0 and ξ(x, t) ξ ∗ (x , t ) = 2δ (x − x ) δ (t − t ), where ξ ∗ denotes the complex conjugate of ξ. To preserve scale invariance we must use a dis√ cretized version of the stochastic forcing ξ(x, t) : (χr + iχi ) / dx dt, where χr and χi are two gaussian distributed fields, which are generated from two uniformly distribuller uted random fields γ1 and γ2 , over the interval (0, 1] using the well known Box–M¨ algorithm χi = −2 log γ1 cos (2πγ2 ) . (4) χr = −2 log γ1 sin (2πγ2 ) ; For the simulation we keep all parameters fixed except for μ, the distance from linear onset, and η, the noise strength. We vary μ from −0.265 until −0.215. Stationary solutions, oscillating pulses with one and two frequencies are included in this range of μ for η = 0. To detect chaotic behavior we use the separation ζ(t) of two nearby states, whose slope gives us a measure of the largest Lyapunov exponent for the whole extended system [64] ζ(t) =

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where the evolution of δA(x, t) comes from a linearization of Eq. (3). Figure 6 shows the separation ζ(t) plotted on a logarithmic scale as a function of t for explosive behaviors (Fig. 6a) and for nonexplosive behaviors (Fig. 6b). Chaotic dynamics leads to an in average positive slope, while nonchaotic dynamics corresponds to an in average negative slope of the separation. Figure 7 summarizes the outcomes of Eq. (3): noisy nonexplosive states, noisy states with explosions, chaotic states without any explosions, and chaotic explosive states [45].

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Fig. 6. Separation ζ(t) plotted on a logarithmic scale as a function of time, μ and η. (a) Explosive behavior for μ = −0.258. The black line stands for chaotic explosive behavior (η = 5.6 · 10−3 ). The gray line corresponds to nonchaotic explosive behavior (η = 3.8 · 10−3 ). (b) Nonexplosive behavior for μ = −0.224. The black line stands for chaotic nonexplosive behavior (η = 3·10−4 ). The gray line corresponds to noisy nonchaotic behavior (η = 2.5·10−4 ).

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We note that for the deterministic case (η = 0), using Poincaré sections and varying the parameter μ, from values corresponding to oscillatory localized solutions until those corresponding to exploding DSs, a narrow range of chaos without explosions has been reported [65]. Noise enhances this small range as it is shown in the phase diagram (see triangles in Fig. 7). For fixed values of μ and increasing η from zero we observe three routes to chaotic localized states with explosions: (a) stationary localized states, noisy states with explosions, chaotic states with explosions; (b) oscillating localized states with one frequency, noisy states with explosions or chaotic states without explosions, chaotic states with explosions; (c) oscillating localized states with two frequencies, chaotic states without explosions, chaotic states with explosions.


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Fig. 8. Phase diagram of possible outcomes starting with localized initial conditions by simulating the two dimensional cubic-quintic Ginzburg-Landau Eq. (2), which includes: solutions with zero amplitude (solid triangles), stationary localized solutions (solid squares), azimuthally symmetric exploding localized solutions (black solid circles), and asymmetric exploding localized solutions (open circles).

2.4 Two classes of exploding dissipative solitons in two spatial dimensions The two dimensional cubic-quintic Ginzburg-Landau Eq. (2) accepts diverse stable localized solutions: stationary pulses, azimuthally symmetric and asymmetric exploding DSs, as it is shown in the phase diagram Fig. 8. The azimuthally symmetric explosions have no analog in one spatial dimension. Symmetric explosions in 1D are only a transient. As usual, to solve an initial value problem, one must specify conditions at t = 0. To get the above mentioned localized solutions the initial conditions must be also localized and overcome a certain threshold in size. Figure 9a shows a subcritical initial condition leading to the stable zero solution (Fig. 9b). While the overcritical initial condition presented in (Fig. 9c) evolves to an asymmetric exploding DS (Fig. 9d). Quasi-one-dimensional (quasi-1D) states are also stable solutions of the two dimensional Eq. (2). This type of functions are localized in one direction and fully extended and space filling in the other direction (see Fig. 9f). A typical non-localized initial condition leading to quasi-1D solutions is shown in (Fig. 9e). Our simulations have been carried out using as a domain a square box of size 50 × 50, 250 × 250 points, dx = dy = 0.2, and dt = 0.01. An explicit fourth-order Runge-Kutta finite differencing numerical method has been implemented. We keep all parameters fixed except for μ, the distance from linear onset, and βi , the nonlinear dispersion. The parameters values are βr = 1.0, γr = −0.1, γi = −0.6, Dr = 0.125, and Di = 0.5. We used periodic boundary conditions. Figure 10 shows snapshots of a stationary localized solution, azimuthally symmetric and asymmetric explosions. In the symmetric case the symmetry is preserved even during the explosion, as it is shown in Fig. 10b. The transition from azimuthally symmetric to asymmetric explosions (see phase diagram Fig. 8) is characterized by an intermittently appearance of blocks of asymmetric explosions. A suitable order parameter characterizing this transition has been introduced in [46]. Figure 11 compares the time evolution of an asymmetric soliton with respect to an azimuthally symmetric soliton. The former moves and tries to visit the whole box in the long time limit. Between two asymmetric explosions the center of mass of the distribution |A(x, y, t)|

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Fig. 9. (a)–(d) μ = −0.1, βi = 1.0. (a) Subcritical localized initial condition occupying 2% of the domain and having 6% of the maximum height of a typical exploding DS. (b) Stable zero solution. (c) Overcritical localized initial condition occupying 2% of the domain and having 13% of the maximum height of a typical exploding DS. (d) Asymmetric exploding soliton occupying 10% of the domain. (e)–(f) μ = −0.5, βi = 1.0. (e) Non-localized initial condition occupying 15% of the domain and having 30% of the maximum height of the quasi-1D solution. (f) Stationary quasi-1D solution occupying 40% of the domain.

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Fig. 10. (a) μ = −0.45, βi = 0.8. Stationary localized solution. (b) μ = −0.45, βi = 0.87. Snapshot of an azimuthally symmetric explosion. The symmetry is preserved even during the explosion. (c) μ = −0.1, βi = 1.0. Snapshot of an asymmetric explosion.

jumps randomly. Thus it describes a random walk, as it is shown in Fig. 12. Note the deterministic character of Eq. (2).

3 Conclusions and perspective In the last section we have given an overview over exploding dissipative solitons as they arise for the cubic-quintic complex Ginzburg-Landau equation. Here we describe some recent related developments and sketch out some future directions. In Ref. [47] we have not only analyzed the existence and stability of quasi one-dimensional localized solutions, but we have also studied their interaction with localized solutions in two spatial dimensions. It turns out that azimuthally symmetric exploding localized solutions and asymmetrically exploding localized solutions show a qualitatively different behavior when interacting with quasi-1 D states. While azimuthally symmetric exploding dissipative solutions can form stable compound


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Fig. 11. (a)–(d) Snapshots of the time evolution of an asymmetric soliton. μ = −0.1, βi = 1.0. (a) T = 0, (b) T = 120, (c) T = 280, (d) T = 480. (e)–(h) Snapshots of the time evolution of an azimuthally symmetric soliton. μ = −0.45, βi = 0.87. (e) T = 0, (f) T = 120, (g) T = 280, (h) T = 480.

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Fig. 12. Random walk of an asymmetric soliton for the first 8 · 105 time units. μ = −0.31, βi = 0.8. Explosions started at the center of the box. This picture has been obtained using a square box of size 100×100, 512×512 points, dx = dy = L/512, and dt = 0.0025. As numerical method a pseudospectral split-step scheme using a fourth order Runge-Kutta algorithm was implemented.

states [47] – we show an example of this type of compound state in Fig. 13b – asymmetric exploding dissipative solitons are annihilated when interacting with quasi-1 D states. This effect is related to the center of mass motion of the asymmetric exploding dissipative solitons leading to dynamic collisions, while azimutally symmetric exploding dissipative solitons show no center of mass motion. For comparison, we have included as Fig. 13a a compound state of a quasi-1D state and a stationary 2 D dissipative soliton. For further detailed analysis we refer to [47]. Burguete et al. [66–68] have recently carried out experiments on a spatially extended fluid layer, which is contained in a rectangular cell, heated at the bottom with a thin heater rail. This quasi-1D extended system develops, as a function of the depth of the fluid layer and the vertical temperature difference, a homogeneous convective

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Fig. 13. (a) μ = −0.45, βi = 0.8. Compound state of a quasi-1 D solution and a 2D stationary pulse. (b) μ = −0.45, βi = 0.85. Compound state of a quasi-1 D solution and an almost azimuthally symmetric exploding DS. (c) μ = −1.24, βi = 1.05. Breathing quasi-1 D with standing waves along the crest.

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Fig. 14. Snapshots of XY plots of the modulus for the breathing quasi-1 D solution shown in Fig. 13c. (a) T = 0, (b) T = 0.905, (c) T = 1.81. The corresponding frequency for the standing wave is ω = 3.46.

pattern (PC), a stationary array of convective cells (ST), oscillatory patterns (TW and ALT) as well as mixed patterns. Our numerical simulations of quasi-1D solutions show some similarities with the above mentioned experimental results. Standing waves along the crest in our quasi-1D solutions from the two-dimensional cubic-quintic complex Ginzburg-Landau equation resemble the behavior observed in ALT patterns (see Figs. 13c and 14). In addition, stationary quasi-1D solutions found in our numerical simulations have their analog in the experimental homogeneous convective patterns (PC) while the stationary array of convective cells (ST) have so far no analog in our numerical picture. For the emergence of secondary instabilities after standing waves we also refer to [69]. Until quite recently all work related to exploding dissipative solitons has been performed for the cubic-quintic complex Ginzburg-Landau equation. To demonstrate that exploding dissipative solitons are a generic phenomenon not restricted to one specific type of envelope equation we have constructed recently a simple reaction-diffusion model with two real variables (for example, the concentration of two chemicals) and polynomial nonlinearities [70]. It turns out that for such a model exploding dissipative solitons not only stably exist, but that they can also propagate over long distances. This observation has been traced back to the lower symmetry of the reaction-diffusion model when compared to the cubic-quintic complex Ginzburg-Landau equation [70]. We note that the transition from non-moving exploding dissipative solitons to propagating dissipative solitons could be induced and controlled by a vorticity parameter (the analogue of detuning in nonlinear optics or the vorticity in fluids).


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There are clearly several directions into which future research on exploding dissipative solitons will go. While exploding dissipative solitons in one spatial dimension have been observed experimentally in nonlinear optics [11] shortly after their theoretical prediction, there are no experimental results on exploding dissipative solitons in two spatial dimensions so far. In addition, it will also be very important to identify experimental systems showing exploding dissipative solitons in one and two spatial dimensions, which can be described by reaction-diffusion type models from chemical reactions or from biology. From a modeling point of view there are at least three directions to go into. One is to find analytic approximation schemes for the various time intervals during a typical cycle of the time evolutions of an exploding dissipative solitons in one and two spatial dimensions. In [43] we were able to capture quantitatively analytically the collapse of the exploding dissipative soliton in one spatial dimension for the cubic-quintic complex Ginzburg-Landau equation. A second avenue of approach is certainly to find exploding dissipative solitons for other types of equations thus demonstrating that they occur for many different systems and under many different circumstances. This includes the study of other prototype equations such as order parameter equations as well as of other basic equations including hydrodynamic equations for simple and complex fluids. A third area of research is the study of the interaction of exploding dissipative solitons in one and two spatial dimensions. These results will have to be contrasted with the available theoretical and experimental work on the interaction of stationary and oscillatory dissipative solitons [5–7, 16–18, 20, 23, 26, 35, 63, 71, 72]. Finally we point out that the field of dissipative solitons (DSs) in general and of exploding DSs in particular is characterized by a large number of open problems and challenges for the mathematics community. Such questions include the regularity necessary for initial conditions and solutions of the cubic-quintic CGL equations, the precise domain of definition of the operator on the right hand side of Eq. (2) as well as the questions whether this operator is pseudomonotone and coercive. Even precise mathematical definitions of the terms dissipative solitons and exploding dissipative solitons will be useful for the mathematics community in the growing field of localized solutions in dissiaptive and driven nonequilibrium systems. O.D. and C.C. wish to thank the support of FONDECYT (Projects No. 1140139 and No. 11121228) and Universidad de los Andes, Chile, through FAI initiatives. HRB thanks the Deutsche Forschungsgemeinschaft (DFG) for partial support of his work.

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