## Abstract

We show the existence of periodic exploding dissipative solitons. These non-chaotic explosions appear when higher-order nonlinear and dispersive effects are added to the complex cubic–quintic Ginzburg–Landau equation modelling soliton transmission lines. This counterintuitive phenomenon is the result of period-halving bifurcations leading to order (periodic explosions), followed by period-doubling bifurcations (or intermittency) leading to chaos (non-periodic explosions).

## 1. Introduction

Experimentally, evidence for soliton explosions was found first by Cundiff *et al.* [1] in Kerr lens mode-locked Ti:sapphire lasers operating in a regime in which the soliton energy suffers dramatic changes. They reported that the intermittent explosions have similar features but are not identical.

Theoretically, explosions were discovered in a continuous model, namely the complex cubic–quintic Ginzburg–Landau equation (CQGLE) [2,3]. However, a more realistic model, considering periodic variations in their parameters, was studied, yielding results close to experiments [1].

In one spatial dimension and by varying only the distance from linear onset (the bifurcation parameter in the context of the onset of instabilities) in the complex CQGLE, we found a transition from stationary pulses to exploding dissipative solitons (DSs) via pulses with only one (rapid) frequency followed by pulses with two frequencies, where one of them is much smaller than the other [4]. By finding an analogue for the Ruelle–Takens route for spatially localized solutions one can see clearly the chaotic nature of explosions. It is remarkable that we did not observe a third frequency mediating chaos [5]. In addition, it has been shown that the appearance of explosions has a signature of intermittency; for instance, the mean value of the time between explosions close to the criticality satisfies a power law [6].

In two spatial dimensions, there are two types of explosions, namely azimuthally symmetric, shown first by Soto-Crespo *et al.* [7], and azimuthally asymmetric explosions [8]. An intriguing issue is that of the type of motion described by the centre of mass of the asymmetric DS. This is still an unsolved problem.

We point out that explosions do not necessarily lead to chaotic behaviour, and nor does chaos lead to explosions. By introducing noise in the complex CQGLE, we found noisy non-chaotic explosive pulses. In addition, it was possible to observe chaotic states without any explosions [9].

By means of a modified complex CQGLE including higher-order nonlinear and dispersive effects, applicable to ultrashort pulses, we show in this article that, besides chaotic exploding pulses, there exist periodic non-chaotic explosions as a result of period-halving bifurcations leading to order (periodic explosions), followed by period-doubling bifurcations (or intermittency) leading to chaos (chaotic explosions).

The production of ultrashort pulses is a very active field in research. This technology finds applications in the areas where a high peak of power is required but the power is turned off before the heat spreads. For instance, biomedical optics, high-speed communications and femto-chemistry, to name a few. By reducing the pulse duration, the laser tool works more precisely and the surrounding material is less exposed to heat. Significant progress has been made in the field of ultrafast lasers over recent decades, ranging from complicated laboratory systems to compact instruments [10]. Mode-locked lasers generate ultra-short pulses by establishing a fixed phase relationship across a broad spectrum of frequencies. Ultrashort pulses are DSs in the sense that they exist because of a complex balance among nonlinearity, dispersion, energy pump and dissipation [11,12].

The paper is organized as follows. In the next section, we describe a model for ultrashort pulses and derive the dimensionless complex CQGLE including effects of third-order dispersion (TOD), self-steepening (SST) and intrapulse Raman scattering (IRS). Section 3 presents our results on transitions from non-periodic to periodic explosions. In §4, we study the influence of noise on periodic explosions, followed in §5 by the Discussion and conclusion.

## 2. Model for ultrashort pulses and the complex Ginzburg–Landau equation including higher-order nonlinear and dispersive effects

Recently, experimental observation of spectral and temporal signatures of soliton explosions in a mode-locked fibre laser has been reported [13]. The system, consisting in an all-normal-dispersion Yb-doped mode-locked fibre laser operating in a transition regime between stable and noise-like emission, has been successfully modelled using a fully realistic iterative cavity map. The considered model for the complex electric field envelope in a comoving frame reads [14]
2.1where corresponds to the gain (or fibre loss), depending on the frequency *ω*, the propagation distance , and *P*, the average energy of the resonating field. *β*_{k} are the dispersion coefficients and *γ*=*ω*_{0}*n*_{2}(*ω*_{0})/(*cA*_{eff}(*ω*_{0})), where *n*_{2}(*ω*_{0}) is the nonlinear refractive index (or nonlinear Kerr parameter) and *A*_{eff} is the effective area of the fibre mode, evaluated at the carrier frequency *ω*_{0}. The time derivative term on the r.h.s. of the above equation is associated with SST and optical shock formation characterized by a time scale *τ*_{shock}∼1/*ω*_{0}. The response function *R*(*T*) includes both instantaneous electronic and delayed Raman contributions.

Equation (2.1) is very general in the sense that it can be used for very short pulses. However, for pulses that are wide enough (for instance approx. 100 fs=0.1 ps or larger for silica), we can make the following approximation [15]: and by neglecting fourth-order dispersion or higher equation (2.1) reduces to 2.2where and .

The numerical results shown in [13] and based on [14] are ultimately related to the mode-locking details. In order to write an equation able to sustain dissipative pulses, it is enough to assume a subcritical shape [16] for , and to consider a term , which might be interpreted as spectral filtering [1]. A large peak power associated with an optical pulse makes it (sometimes) necessary to replace *γ* by , where *γ*_{1} is a saturation parameter.

In order to obtain a normalized equation for the complex electric field envelope, we consider the physical quantities *P*_{0}, *L*_{D} and *T*_{0}, which correspond to the peak power, the dispersion length and the width of the incident pulse, respectively. By introducing the dimensionless variables (*ψ*,*z*,*τ*), we obtain
2.3where and .

Using the scaling (2.3) in equation (2.2), we get the dimensionless complex CQGLE including higher-order nonlinear and dispersive effects (H.O.E):
2.4where *δ*_{3}≡*β*_{3}/6*T*_{0}|*β*_{2}|,*s*≡1/*ω*_{0}*T*_{0},*τ*_{R}≡*T*_{R}/*T*_{0}, *δ* accounts for the linear gain/loss, *β* stands for the spectral filtering, *ϵ* and *μ* account for the nonlinear gain/absorption, and *ν* corresponds to a saturation higher-order term related to the intensity-dependent refractive index. The parameter *ϵ* is a suitable control parameter as it is related to the pumping power. The parameters *δ*_{3}, *s* and *τ*_{R} represent, respectively, the effects of TOD, SST and IRS.

The l.h.s. of equation (2.4) contains conservative terms. In fact, except for the quintic-order term we recover the nonlinear Schrödinger equation. While on the r.h.s., the terms related to *δ*, *ϵ*, *μ* and *β* are dissipative, the H.O.E. include only conservative terms. In particular, it is straightforward to check that does not change its value because of TOD, SST or IRS.

Tian *et al.* [17] found first that nonlinear gradient terms result in dramatic changes in the soliton behaviour leading to fixed-shape solitons. Later, Latas & Ferreira [18–20] found that explosions can be controlled if these higher-order effects are properly conjugated two by two. Evolution of pulses in the presence of higher-order effects has been studied by Facão & Carvalho [21,22]. Recently, Cartes & Descalzi [23] have shown that the transition between explosive and regular behaviour can be characterized by a transcritical bifurcation controlled by the SST parameter. Much earlier, the effect of nonlinear gradient terms on localized states in a complex CQGLE was studied by Deissler & Brand [24], showing that these terms lead to asymmetries and reduction of the speed of the localized solutions.

## 3. Periodic and non-periodic explosions

This section shows the results obtained by simulating numerically equation (2.4), which is the normalized complex CQGLE including higher-order nonlinear and dispersive effects. The parameters (*δ*_{3},*s*,*τ*_{R}) play a role only for ultrashort pulses (*T*_{0}<1 ps). Otherwise, they are negligible. Taking into account that *T*_{R}∼3 fs [15] (valid for silica), we can estimate *τ*_{R}∼0.03. On the other hand, *s*=1/*ω*_{0}*T*_{0}∼1/*N*, where *N* is the number of cycles involved by the envelope *ψ*. Typically, *s* and *δ*_{3} are smaller than *τ*_{R}.

For the simulations of equation (2.4), we keep all parameters fixed except for *ε*, which plays the role of a control parameter directly related to the pumping power: *δ*=−0.1,*β*=0.125, *μ*=−0.1,*ν*=0.6. Simulations have been carried out using a box of size *N*=8192 points, d*t*=0.01 with *T*=81.92, d*z*=0.004, along a pseudo-spectral split-step method to compute the differential operator and a fourth-order Runge–Kutta scheme for the integration in *z*. In this article, we consider two cases: (I) *δ*_{3}=0.016,*s*=0.009 and *τ*_{R}=0.032, and (II) *δ*_{3}=0.018,*s*=0.012 and *τ*_{R}=0.09206.

In figure 1, we plot, for the case (I), the energy for three different values of *ε*. Depending on *ε*, we can observe a behaviour with two frequencies (figure 1*a*), four frequencies (figure 1*b*) and chaos (figure 1*c*). The maximum values of , as a function of *ε* represent a logistic map displaying a complex picture, as shown in figure 2*a*. A first inspection of this figure gives us an indication of where chaos can be structurally unstable, that is, arbitrarily small changes of a parameter (not necessarily *ε*) can lead to periodic behaviour. In addition, we observe that there are ‘large’ windows with periodic dynamics. The same results have been obtained for many different values of (*δ*_{3},*s*,*τ*_{R}). The windows around *ε*=1.0 and *ε*=1.02 have been plotted with higher resolution in figure 2*b*,*c*, respectively. In both cases, period-halving bifurcations followed by period-doubling bifurcations are leading to order (periodic explosions) and chaos (non-periodic explosions), respectively. Recently, Ott and co-workers [25] addressed the following important question: how certain is it that an attractor is chaotic?

In figure 3, we plot the peak of the amplitude |*ψ*|_{peak} versus the energy *Q* for two frequencies (figure 3*a*), four frequencies (figure 3*b*) and chaos (figure 3*c*), corresponding to the plots shown in figure 1. From these plots, one can see easily the difference in behaviour between periodic and chaotic states. In contrast, the evolution of |*ψ*| in a *τ*−*z* plot, as shown in figure 4, does not offer a clear means to distinguish between order and chaos.

After a period, *ζ*,|*ψ*(*t*,*z*_{0}+*ζ*)| coincides exactly with |*ψ*(*t*,*z*_{0})| even on a logarithmic scale, as can be seen in figure 5. We also studied the persistence of the periodic behaviour for very long propagation distances, to be sure that this behaviour is not a very slow transient.

For the case (II), where *δ*_{3}=0.018,*s*=0.012 and *τ*_{R}=0.09206, we can obtain its corresponding logistic map for covering a large range of *ε* (figure 6). Their main characteristic features are the same as those already described for the case (I): chaos can be structurally unstable, fractal and have windows for periodic explosions. However, figure 7 (a zoom of figure 6 around *ε*=1.0) shows period-halving bifurcations leading to order (periodic explosions) followed by intermittency leading to chaos (non-periodic explosions). Compare this with case (I), where order was followed by period-doubling leading to chaos. The time series for case (II) before and after intermittency is shown in figure 8.

## 4. Influence of noise on periodic explosions

In order to test the stability under small perturbations of the periodic explosive solitons, we measure the effects of white noise on the dynamics by using equation (2.4) with additive noise so the complete stochastic system becomes
4.1where *η* is the variance which is related to the standard deviation *σ* by the relation *η*=*σ*^{2}, and *ξ*(*x*,*t*) is the stochastic forcing with the following properties: 〈*ξ*〉=0 and its correlation is 〈*ξ*(*x*,*t*)*ξ**(*x*′,*t*′)〉=2*δ*(*x*−*x*′)*δ*(*t*−*t*′), where *ξ** corresponds to the complex conjugate of *ξ*.

To solve numerically equation (4.1), the discretized version of the stochastic force has to be scaled by a factor , and the real and imaginary parts of *ξ* are Gaussian fields with zero mean and unit variance, generated from two uniformly distributed random fields on the interval (0,1], λ_{1} and λ_{2}, by using the Box–Müller algorithm
and

The effects of a very small amount of noise, *η*=10^{−7}, can be observed in figure 9*a*. It is evident that the general shape still preserves the main features shown in figure 7 and two of the period-doubling transitions. The dynamics of the explosive structure is mainly periodic, as can be seen in figure 10*a*, but there are some qualitative differences (see figure 10 inset). If noise strength is increased, to *η*=5×10^{−7}, the logistic map (figure 9*b*) shows almost no periodic features. Nevertheless, |*ψ*|_{peak} as a function of *Q* is nearly periodic except for some stochastic details, as shown in figure 10*b*. Finally, if noise strength is further increased, to a value of around *η*=10^{−5}, all the periodic features are lost and the stochastic explosions are indistinguishable from their chaotic counterpart.

## 5. Discussion and conclusion

By assuming very short pulses (but wide enough to contain many cycles), from a general model, which has been successfully describing explosions in a mode-locked fibre laser, we derive the complex CQGLE including higher-order nonlinear and dispersive effects. By plotting the maximum values of the ‘energy’ as a function of a control parameter related to the pumping power, we obtain a logistic map with characteristic features: chaos can be structurally unstable, fractal and have windows for periodic explosions.

In this article, we have shown the existence of periodic exploding DSs. This counterintuitive phenomenon is the result of period-halving bifurcations leading to order (periodic explosions), followed by period-doubling bifurcations (or intermittency) leading to chaos (non-periodic explosions).

In [2,3] it has already been shown that exploding DSs exist over a substantial range of values of the quintic refractive index and of the value of the cubic destabilizing coupling in the plain CQGLE. Later, it was reported in [4,5] that exploding dissipative solitons arise from modulated oscillatory solitons characterized by two frequencies leading to an analogue of the Ruelle–Takens route for spatially localized solutions. Power spectra of explosions show broad-band low-frequency noise characteristics for low-dimensional chaotic systems. In [6], the distribution of time between explosions was studied. Its distribution is far from exponential and suggests that there is some memory effect induced by the structure of the attractor. In contrast, periodic explosions exist only in the complex CQGLE when higher-order nonlinear and dispersive effects are included. They have a delta-shaped distribution for the time between explosions. Then they are not chaotic (by definition).

Natural candidates to test our predictions are Kerr lens mode-locked Ti:sapphire lasers and Yb-doped mode-locked fibre lasers.

## Authors' contributions

Both authors contributed equally to this paper by means of substantial contributions to the 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

We received no funding for this study.

## Acknowledgements

C.C. and O.D. wish to acknowledge the support of FONDECYT (projects nos. 11121228 and 1140139) and Universidad de los Andes through FAI initiatives.

## Footnotes

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

- Accepted July 17, 2015.

- © 2015 The Author(s)