In the last decades, rapid progress in modern nonlinear science was marked by the development of the concept of dissipative soliton (DS). This concept is highly useful in many different fields of science ranging from field theory, optics, and condensed matter physics to biology, medicine, and even sociology. This chapter aims to present a DS appearance from random fluctuations, development, and growth, the formation of the nontrivial internal structure of mature DS and its breakup, in other words, a full life cycle of DS as a self-organized object. Our extensive numerical simulations of the generalized cubic-quintic nonlinear Ginzburg-Landau equation, which models, in particular, dynamics of mode-locked fiber lasers, demonstrate a close analogy between the properties of DS and the general properties of turbulent and chaotic systems. In particular, we show a disintegration of DS into a noncoherent (or partially coherent) multisoliton complex. Thus, a DS can be interpreted as a complex of nonlinearly coupled coherent “internal modes” that allows developing the kinetic and thermodynamic theory of the nonequilibrious dissipative phenomena. Also, we demonstrate an improvement of DS integrity and, as a result, its disintegration suppression due to noninstantaneous nonlinearity caused by the stimulated Raman scattering. This effect leads to an appearance of a new coherent structure, namely, a dissipative Raman soliton.
- optical turbulence
- dissipative solitons
- chaos in nonlinear optical systems
- generalized cubic-quintic nonlinear Ginzburg-Landau equation
- dissipative Raman soliton
Coherent and partially coherent structures emerging in nonlinear systems far from the thermodynamic equilibrium play an important role in different research areas ranging from hydrodynamics and plasma physics to biophysics and sociology. Nontrivial dynamics of such structures including chaos and turbulence is a challenge for modern nonlinear science and one may assume that “the problem of turbulence is one of the central problems in theoretical physics” . The reasonable approach to this issue, which can translate some contra-intuitive and obscure ideas in this area into explicit and verifiable concepts, is a realization of simpler dynamics in quite different material context. Such an approach can be named
In this work, we conjecture a spectacular analogy between the spectral structures of DS and strong Langmuir turbulence. Such close relation leads to chaotization of DS dynamics with the energy growth. This analogy is deepened by analysis of energy flows inside DS so that a DS can be represented as a “glass of boiling water” or, mathematically, as an ensemble of interacting quasi-particles or “nonlinear modes.” The phase decoupling of these “modes” leads to turbulence or DS dissolving. Such a representation open the door for building the
2. Analogy between DS and turbulence
The phenomenon of turbulence appears in many areas of our experience ranging from atmospheric and oceanic rogue events, aero- and hydrodynamics, optics to cardiology and neurophysiology [1, 5, 7, 8, 9, 10, 11, 12, 13]. Such a broad class of phenomena cannot be grasped by some single and simple model. However, there are some comparatively simple equations which allow describing an extremely broad class of phenomena. It is possible that the most known one is the famous nonlinear Schrödinger equation (NSE) which describes an evolution of slowly varying wave in a nonlinear medium and can be considered as a “metaphoric” simulation tool for a study of nonlinear phenomena far from equilibrium [14, 15]:
The dimensionality of this equation is relative: the evolutional coordinate can be a time
The notion of turbulence is fuzzy in some sense. Here, the turbulence will be treated as a phenomenon related to the excitation of a sufficiently large number of degrees of freedom that causes a loss of their mutual phase information . As a consequence, a wave package decouples into a set of individual modes (“particles”) which interaction can be described in the framework of kinetic theory as many-particle collisions in Bose-gas. In other words, as some
Thus, we come to a
Such an equation becomes nontrivial in a dissipative environment [17, 18]. A simple generalization of NSE (1) taking into account the dissipative effects includes a saturable gain (energy “source”)
Eqs. (1) and (4) called the generalized complex nonlinear Ginzburg-Landau equation have the strongly localized (in
Dissipation adds new bounds on the soliton parameters and fixes them so that one may say that the DS lives in “the heart of nonlinear world” (“bodhisattva,” Figure 2).
The mutual balance of dispersion and phase nonlinearity remains a crucial factor for DS formation, but its physical meaning differs substantially from that for nondissipative soliton. The crucial factor here is a
However, sole dispersive balance is not sufficient for the DS stability. The spectral dissipation ∼
The key feature of DS is its nontrivial internal structure revealing itself in the phase inhomogeneity4 and the internal energy redistribution (
The third term in RHS of Eq. (5) is phase-sensitive and, thus, there is an energy flow from DS center, where spectral components with minimal relative frequencies are located, to the DS wings, where frequency components with maximum relative frequencies are located (Figure 5). Here, energy dissipates. Such nontrivial internal “life” of DS intensifies with the growth of phase inhomogeneity Θ. Simultaneously, DS becomes an energy scalable coherent concentrate with the energy (“concentrate mass”) ∝Θ .
As a result of phase inhomogeneity and intensive internal energy flows, the internal coherence of DS can become partially broken. Then, DS splits into partially coherent “internal modes” which interact with each other as the independent “sub-solitons.” [29, 30, 31] Thus, DS becomes a strongly localized “cloud” of interacting “quasi-particles” or “glass of boiling water” (Figure 6).
These figures demonstrate an affinity between the structures of DS and turbulence [8, 23]. Both spectral structures are defined by dispersion relations: between soliton and dispersive waves for the former and Langmuir dispersion relation for the latter (Figure 7). Secondly, both high-energy DS and turbulence are characterized by spectral condensation at zero frequency (wavenumber) with subsequent scattering to higher frequencies confined by cut-off at ±Δ.
Such an analogy between DS and turbulence opens a door for building the kinetic and quantum [5, 32, 33, 34] theory of open (dissipative) semi-coherent structures which mimics, in particular, a quantum Bose-Einstein condensate in a dissipative environment.
3. Transition to a DS turbulence
The mechanism of transition to turbulence for DS can be associated with the time/spectral duality (Figure 8). When the energy increases (i.e.,
More close insight into this mechanism can be provided by the adiabatic theory of DS in spectral domain presented in . As was shown, the DS spectrum can be expressed as follows:
Since the “chemical potential” Ξ2 decays with the energy growth (Figure 8), a system tends to the state of “soliton gas”  with the characteristic “soliton size” ∝1/Δ. Thereby, a coherent “condensate” with minimum entropy becomes a state of the decomposed “quasi-particles” with the chaotically modulated powers because the required entropy growth is provided by such modulation7 . Thus, the energy growth (i.e., the growth of “condensate mass” ∝1/Ξ) leads to extra-sensitivity to
The example of such “DS decomposition” through a turbulence is shown in Figure 9. This figure is obtained by numerical simulation of Eqs. (1) and (4) with taking into account of the gain saturation in the form of
An analysis of turbulent DS demonstrates its complicate internal structure which can be interpreted as the complex of strongly interacting bright, dark, and gray DSs on a finite but strongly self-localized background concentrating almost the entire part of the energy.8 In some sense, an appearance of DS turbulence resembles the laminar-turbulent transition in a fiber laser when a macroscopically coherent field (Ξ → 0) becomes chaotically self-modulated .
Nevertheless, such a scenario is not unique. The turbulent dynamics can result from the strong interaction between “individual” DSs forming a “soliton gas” or turbulent “soliton cluster” (Figure 10) [1, 5, 47]. Interaction of such cluster with a low-intensity background field can result in permanent radiation or absorption of DSs in the form of so-called “soliton rains” [48, 49].
Separately, one may note the chaotization of DS dynamics caused by resonant interaction with the dispersive waves in the presence of higher-order corrections to the dispersion term in Eq. (1). In this case, the collisions between DS and dispersive wave, which radiates by it, results in a chaotic dynamic preserving, nevertheless the DS integrity (Figure 11) .
4. Coherence of DS in the presence of nonlinearity with nonlocal/noninstantaneous response
The simulations demonstrate [6, 35] that SRS suppresses the DS turbulence for the sufficiently large dispersions
DRS is characterized by large chirp Θ and frequency down-shift. The last results from intra-pulse SRS which is possible due to a broad spectrum, which is a common characteristic of DS and results from its large Θ. A sole DRS develops with growing
Another spectacular manifestation of the effect of a noninstantaneous nonlinearity on an incoherent field is an appearance of the spectral incoherent solitons (SIS) . The spectrally localized soliton-like structures appear without any time-localization due to the causality property inherent to SRS so that a field cannot reach thermal equilibrium . Formally, the corresponding evolution equation in the Langmuir turbulence limit has soliton-like solutions in the spectral domain [55, 56]. As a result, such structure localized in spectral domain possesses main properties of solitons including the property of elastic scattering.
The problem of DS coherence, chaotic, and turbulent dynamics has been outlined. A nontrivial internal structure of DS caused by it intensive energy exchange with dissipative environment allows conjecturing a close analogy with turbulent structure forming far from equilibrium. The existence of long-range correlation scale provides the DS energy scaling (or mass scaling for Bose-Einstein condensate). However, such “macroscopic” scaling is provided by strong phase inhomogeneity (chirp) so that internal coherence of DS defined by short-range correlation scale breaks and DS becomes a “cloud” of interacting “quasi-particles” or “glass of boiling water.” Such structure is very sensitive to perturbation of even quantum level. Such extra-sensitivity combines macro- and micro-scales that raises an issue of the quantum theory of the macroscopic coherent, partially, and incoherent dissipative structures.
In the context of this work, such DS “decomposition” leads to turbulent dynamics and DS fragmentation. In particular, interactions inside such “soliton cluster” can result in the rogue waves’ formation. An additional source of soliton destabilization is resonant interaction with a dispersive wave that results in chaotization of dynamics and formation of “soliton rains.”
Nonlinearity with noninstanteneous response (e.g., SRS) leads to new interesting effect. In particular, SRS suppresses the DS turbulence due to the formation of DS + DRS pairs or sole DRS. Although DRS is turbulence-free within a broad parametric range, it has a perturbed anti-Stokes component, which causes chaotic vibrations of DRS parameters.
Another and spectacular manifestation of the noninstantaneous response of nonlinearity is the formation of SIS. This structure is a soliton in the spectral domain but incoherent and delocalized in the time domain.
The above-considered phenomena and conjectures are of interest in the context of the development of approaches to the self-consistent theory of nonequilibrium dissipative structures in classical and quantum aspects, which would use the optical DSs as a testbed.
The author acknowledges the support from Austrian Science Fund (FWF Project No. P24916-N27). Computational results have been achieved using the Vienna Scientific Cluster (VSC).
- Further, namely one-dimensional (d = 1) systems will be under consideration that is a quite precise approximation for solid-state and fiber laser dynamics .
- One has to remind the x ↔ t and k ↔ ω dualities in Eq. (1).
- More precise analysis  gives the conditions of asymptotical stability: αγ βκ ≤ 1 / 3 if E → ∞ 1 / 2 if E → 0 , where E is a DS energy.
- The measure of this inhomogeneity is a so-called chirp Θ ∝ ∂ 2 arg Ψ ∂ t 2 .
- The value ∝1/Ξ can be treated as a measure of “long-range” correlation scale.
- The value ∝1/Δ can be treated as a measure of “short-range” correlation scale.
- Here, one may draw an analogy with Hamiltonian systems, where the gradient of field is a measure of the amount of fluctuations .
- One has to distinguish such a structure from the breather-like structures on a continuous-wave background. Such structures can demonstrate chaotic and rogue waves dynamics, as well (e.g., see [43, 44]).