Dissipative soliton

Dissipative solitons (DSs) are stable solitary localized structures that arise in nonlinear spatially extended dissipative systems because of self-organization. They can be considered as an extension of the classical soliton concept in conservative systems. An alternative terminology includes autosolitons, spots and pulses.

Apart from aspects similar to the behavior of classical particles like the formation of bound states, DSs exhibit entirely nonclassical behavior – e.g. scattering, generation and annihilation – all without the constraints of energy or momentum conservation. The excitation of internal degrees of freedom may result in a dynamically stabilized intrinsic speed, or periodic oscillations of the shape.

Historical development

Origin of the soliton concept

DSs have been experimentally observed for a long time. In 1831, Faraday ^[1] saw solitary localized structures in a vibrating layer of fine powder. Helmholtz^[2] measured the propagation velocity of nerve pulses in 1850. In 1902, Lehmann^[3] found the formation of localized anode spots in long gas-discharge tubes. Nevertheless, the term "soliton" has originally developed in a different context. The starting point was the experimental detection of "solitary water waves" by Russell in 1834^[4]. These observations initiated the theoretical work of Rayleigh^[5] and Boussinesq^[6] around 1870 which finally lead to the approximate description of such waves by what is known today as the (conservative) KdV equation by Korteweg and de Vries^[7] (1895). On this background the term "soliton" has been coined by Zabusky and Kruskal^[8] in 1965. These authors investigated certain well localised solitary solutions of the KdV equation and named these objects solitons. Among other things they demonstrated that in 1-dimensional space solitons exist e.g. in the form of two unidirectionally propagating pulses with different size and speed having the remarkable property that number, shape and size are the same before and after collision.

Gardner at al.^[9] introduced the inverse scattering technique for solving the KdV equation and proved that this equation is completely integrable. In 1972 Zakharov and Shabat^[10] found another integrable equation and finally it turned out that the inverse scattering technique can be applied successfully to a whole class of equations (e.g. the Nonlinear Schrödinger and the Sine-Gordon equation). From 1965 up to about common agreement to reserve the term soliton to pulse-like solitary solutions of conservative nonlinear partial differential equations that can be solved by using the inverse scattering technique.

Weakly and strongly dissipative systems

With increasing knowledge on classical solitons, possible technical applicability came into perspective, with the most promising one at present being the transmission of optical solitons via glass fibers for the purpose of data transmission. In contrast to systems with purely classical behavior, solitons in fibers dissipate energy and this cannot be neglected on an intermediate and long time scale. Nevertheless the concept of a classical soliton can still be used in the sense that on a short time scale dissipation of energy can be neglected. On an intermediate time scale one has to take into account small energy losses as a perturbation and on a long scale the amplitude of the soliton will decay and finally vanish^[11].

There are however various types of systems which are capable of producing solitary structures and in which dissipation plays an essential role for their formation and stabilization. Although research on certain types of these DSs has been carried out for a long time (compare e.g. the works on nerve pulses culminating in the work of Hodgkin and Huxley^[12] in 1952), since about 1990 the amount on research has significantly increased. Possible reasons are improved experimental devices and analytical techniques as well as the availability of more powerful computers for numerical computations. Nowadays, it is common agreement to use the term DSs for solitary structures in strongly dissipative systems.

Experimental observations of DSs

Today, DSs can be found in various types of different experimental set-ups. Examples include

- Gas-discharge systems: plasmas confined in a discharge space which often has a lateral extension large compared to the main discharge length. DSs arise as current filaments between the electrodes and were found in dc systems with high-ohmic barrier^[13] , ac systems with dielectric barrier^[14], as anode spots^[15] as well as in an obstructed discharge with metallic electrodes^[16].

• DSs experimentally observed in planar dc gas-discharge systems with high-ohmic barrier
•  
  Without oscillatory tails
•  
  With oscillatory tails

- Semiconductor systems: similar to gas-discharges, however instead of a gas semiconductor material is sandwiched between two planar or spherical electrodes. Set-ups include Si and GaAs pin diodes^[17], n-GaAs^[18], Si p⁺-n⁺-p-n^-[19] and ZnS:Mn structures^[20] .

- Nonlinear optical systems: a light beam of high intensity interacts with a nonlinear medium, typically the medium reacts on rather slow time scales compared to the beam propagation time. Often, the output is fed back into the input system via single-mirror feedback or a feedback loop. DSs may arise as bright spots in two-dimensional plane orthogonal to the beam propagation direction, one may however also exploit other effects like polarization. DSs have been observed for saturable absorbers^[21], degenerate optical parametric oscillators (DOPOs)^[22], liquid crystal light valves (LCLVs)^[23], alkali vapor systems ^[24], photorefractive media^[25] and semiconductor microresonators^[26] .

Chemical systems: realized either as one- and two-dimensional reactors or via catalytic surfaces, DSs appear as (often as propagating) pulses of increased concentration or temperature. Typical reactions are the Belousov-Zhabotinsky_reaction^[27], the ferrocyanide-iodate-sulphite as well as the oxidation of hydrogen^[28], CO^[29] or iron^[30]. Nerve pulses^[31] also belong to this class of systems.

- Vibrated media: vertically shaken granular media^[32], colloidal suspensions^[33] or Newtonian fluids^[34] produce harmonically or sub-harmonically oscillating heaps of material, which are usually called oscillons.

- Hydrodynamic systems: the most prominent realization of DSs are domains of convection rolls on a conducting background state in binary liquids^[35]. Another example is film dragging in a rotating cylindric pipe filled with oil^[36].

- Electrical networks: large one- or two-dimensional arrays of coupled cells with a nonlinear current-voltage characteristic^[37], DSs are characterized by a locally increased current through the cells.

Remarkably enough, phenomenologically the dynamics of the DSs in many of the above systems is similar in spite of the microscopic differences. Typical observations are (intrinsic) propagation, scattering, formation of bound states and clusters, drift in gradients, interpenetration, generation and annihilation as well as higher instabilities.

Theoretical description of DSs

Most systems showing DSs are described by nonlinear partial differential equations, sometimes also discrete equations or cellular automata are used. Up to now, modeling from first principles followed by a quantitative comparison of experiment and theory as been performed only rarely and sometimes also poses severe problems due to to large discrepancies between microscopic and macroscopic time and space scales. Often simplified prototype models are investigated which reflect the essential physical processes in a larger class of experimental systems. Among these are

- Reaction-diffusion systems, used for chemical systems, gas-discharges and semiconductors^[38]. The evolution of the state vector ${\displaystyle {\boldsymbol {q}}({\boldsymbol {x}},t)}$  describing the concentration of the different reagents is determined by diffusion as well as local reactions:

${\displaystyle \partial _{t}{\boldsymbol {q}}={\underline {\boldsymbol {D}}}\Delta {\boldsymbol {q}}+{\boldsymbol {R}}({\boldsymbol {q}}).}$ 

A frequently encountered example is the two-component Fitzhugh-Nagumo-type activator-inhibitor system

${\displaystyle \left({\begin{array}{c}\tau _{u}\partial _{t}u\\\tau _{v}\partial _{t}v\end{array}}\right)=\left({\begin{array}{cc}d_{u}^{2}&0\\0&d_{v}^{2}\end{array}}\right)\left({\begin{array}{c}\Delta u\\\Delta v\end{array}}\right)+\left({\begin{array}{c}\lambda u-u^{3}-\kappa _{3}v+\kappa _{1}\\u-v\end{array}}\right).}$ 

Stationary DSs are generated by production of material in the center of the DSs, diffusive transport into the tails and depletion of material in the tails. A propagating pulse arises from production in the leading and depletion in the trailing end^[39]. Among others, one finds also periodic oscillations of DSs^[40], bound states^[41], collisions, merging, generation and annihilation^[42].

- Ginzburg-Landau type systems for a complex scalar ${\displaystyle q({\boldsymbol {x}},t)}$  used to describe e.g. nonlinear optical systems, plasmas, Bose-Einstein condensation, liquid crystals and granular media^[43]. A frequently found example is the cubic-quintic subcritical Ginzburg-Landau equation

${\displaystyle \partial _{t}q=(d_{r}+id_{i})\Delta q+l_{r}A+(c_{r}+ic_{i})|q|^{2}q+(q_{r}+iq_{i})|q|^{4}q.}$ 

An alternative formulation is given by considering the energy ${\displaystyle \rho =q^{2}}$  for which one may derive the continuity equation

${\displaystyle \partial _{t}\rho +\nabla \cdot {\boldsymbol {m}}=S=d_{r}(q\Delta q^{\ast }+q^{\ast }\Delta q)+2l_{r}\rho +2c_{r}\rho ^{2}+2q_{r}\rho ^{3}\quad {\text{with}}\quad {\boldsymbol {m}}=2d_{i}{\text{Im}}(q^{\ast }\nabla q).}$ 

One can thereby show that energy is produced in the flanks of the DSs and transported to the center and potentially to the tails where it is depleted. Dynamical phenomena include propagating DSs in 1d^[44], propagating clusters in 2d^[45], bound states, vortex solitons^[46] as well as "exploding DSs"^[47].

- Swift-Hohenberg equation: Used e.g. in nonlinear optics, granular media dynamics of flames or electroconvection, the Swift-Hohenberg can be considered as an extension of the Ginzburg-Landau equation. It can be written as

${\displaystyle \partial _{t}q=(s_{r}+is_{i})\Delta ^{2}q+(d_{r}+id_{i})\Delta q+l_{r}q+(c_{r}+ic_{i})|q|^{2}q+(q_{r}+iq_{i})|q|^{4}q.}$ 

For ${\displaystyle d_{r}>0}$  one essentially has the same mechanisms as in the Ginzburg-Landau equation^[48]. For ${\displaystyle d_{r}<0}$ , in the real Swift-Hohenberg equation one finds bistability between homogeneous states and Turing patterns. DSs are stationary localized Turing domains on the homogeneous background^[49]. This also holds for the complex Swift-Hohenberg equations, however here also propagating DSs as well as interaction phenomena are possible, observations include merging and interpenetration^[50].

Particle properties and universality

DSs in many different systems show universal particle-like properties. To understand and describe the latter, one may try to derive "particle equations" for slowly varying order parameters like position, velocity or amplitude of the DSs by adiabatically eliminating all fast variables in the field description. This technique is known from linear systems, however mathematical problems arise from the nonlinear models due to a coupling of fast and slow modes^[51].

Similar to low-dimensional dynamic systems, for supercritical bifurcations of stationary DSs one finds characteristic normal forms essentially depending on the symmetries of the system. E.g., for a transition from a symmetric stationary to an intrinsically propagating DS one finds the Pitchfork normal form

${\displaystyle {\dot {\boldsymbol {v}}}=(\sigma -\sigma _{0}){\boldsymbol {v}}-|{\boldsymbol {v}}|^{2}{\boldsymbol {v}}}$ 

for the velocity ${\displaystyle {\boldsymbol {v}}}$  of the DS^[52], here ${\displaystyle \sigma }$  represents the bifurcation parameter and ${\displaystyle \sigma _{0}}$  the bifurcation point. For a bifurcation to a "breathing" DS, one finds the Hopf normal form

${\displaystyle {\dot {A}}=(\sigma -\sigma _{0})A-|A|^{2}A}$ 

for the amplitude ${\displaystyle A}$  of the oscillationCite error: A <ref> tag is missing the closing </ref> (see the help page).. In this way, a comparison between experiment and theory is facilitated^[53], ^[54]. Note that the above problems do not arise for classical solitons as inverse scattering theory yields complete analytical solutions.

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