Dissipative solitons are stable solitary localized
structures that arise in nonlinear spatially extended
dissipative system

Dissipative system

A dissipative system is a thermodynamically open system which is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter....

s due to mechanisms of self-organization

Self-organization

Self-organization is the process where a structure or pattern appears in a system without a central authority or external element imposing it through planning...

.
They can be considered as an extension of the classical
soliton

Soliton

In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...

 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

Degrees of freedom (physics and chemistry)

A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

 may result in a dynamically stabilized intrinsic
speed, or periodic oscillations of the shape.

Origin of the soliton concept

DSs have been experimentally observed for a long time.
Helmholtz measured the propagation velocity of nerve pulses in
1850. In 1902, Lehmann

Otto Lehmann

Otto Lehmann was a German physicist and "father" of liquid crystal technology.-Life:...

 found the formation of localized anode
spots in long gas-discharge tubes. Nevertheless, the term
"soliton" was originally developed in a different context. The
starting point was the experimental detection of "solitary
water waves" by Russell

John Scott Russell

John Scott Russell was a Scottish naval engineer who built the Great Eastern in collaboration with Isambard Kingdom Brunel, and made the discovery that gave birth to the modern study of solitons.-Personal life:John Scott Russell was born John Russell on 9 May 1808 in Parkhead, Glasgow, the son of...

 in 1834.
These observations initiated the theoretical work of
Rayleigh

John Strutt, 3rd Baron Rayleigh

John William Strutt, 3rd Baron Rayleigh, OM was an English physicist who, with William Ramsay, discovered the element argon, an achievement for which he earned the Nobel Prize for Physics in 1904...

 and Boussinesq around
1870, which finally led to the approximate description of such
waves by Korteweg and de Vries in 1895; that description is known today as the (conservative)
KdV equation.

On this background the term "soliton

Soliton

In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...

" was
coined by Zabusky

Norman Zabusky

Norman J. Zabusky is an American physicist, who is noted for the discovery of the soliton in the Korteweg–de Vries equation, in work completed with Martin Kruskal. This result early in his career was followed by an extensive body of work in computational fluid dynamics, which led him more...

 and Kruskal 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 and exhibiting the
remarkable property that number, shape and size are the same
before and after collision.

Gardner at al. introduced the
inverse scattering technique

Inverse scattering transform

In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. It is one of the most important developments in mathematical physics in the past 40 years...

for solving the KdV equation and proved that this equation is
completely integrable

Integrability

Integrability may refer to:* Riemann integrability; see Riemann integral* Lebesgue integrability; see Lebesgue integral* Darboux integrability; see Darboux integral* System integration * Interoperability...

. In 1972 Zakharov

Vladimir E. Zakharov

Vladimir Evgen'evich Zakharov is a Soviet and Russian mathematician and physicist. He is currently Regents' Professor of mathematics at The University of Arizona and director of the Mathematical Physics Sector at the Lebedev Physical Institute...

 and
Shabat 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

Nonlinear Schrödinger equation

In theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödinger's equation. It is a classical field equation with applications to optics and water waves. Unlike the Schrödinger equation, it never describes the time evolution of a quantum state...

 and
Sine-Gordon

Sine-Gordon equation

The sine–Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally considered in the nineteenth century in the course of study of surfaces of constant negative...

 equations). From 1965
up to about 1975, a common agreement was reached: 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 of classical solitons, possible
technical applicability came into perspective, with the most
promising one at present being the transmission of optical
solitons via glass fibers

Fiberglass

Glass fiber is a material consisting of numerous extremely fine fibers of glass.Glassmakers throughout history have experimented with glass fibers, but mass manufacture of glass fiber was only made possible with the invention of finer machine tooling...

 for the purpose of
data transmission

Data transmission

Data transmission, digital transmission, or digital communications is the physical transfer of data over a point-to-point or point-to-multipoint communication channel. Examples of such channels are copper wires, optical fibres, wireless communication channels, and storage media...

. 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 small energy losses into account as a
perturbation, and on a long scale the amplitude of the soliton
will decay and finally vanish.

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 (for example, see the research on nerve pulses culminating
in the work of Hodgkin and Huxley in 1952), since
1990 the amount of research has significantly increased (see e.g.

)
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 to use the term dissipative solitons for solitary structures in
strongly dissipative systems.

Experimental observations of DSs

Today, DSs can be found in many different
experimental set-ups. Examples include

• Gas-discharge systems
  Gas-discharge lamp
  Gas-discharge lamps are a family of artificial light sources that generate light by sending an electrical discharge through an ionized gas, i.e. a plasma. The character of the gas discharge critically depends on the frequency or modulation of the current: see the entry on a frequency classification...
  
  : plasmas
  Plasma (physics)
  In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...
  
   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 a high-ohmic barrier, AC systems with a dielectric barrier, and as anode spots, as well as in an obstructed discharge with metallic electrodes.

• Semiconductor
  Semiconductor
  A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
  
   systems: these are 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
  PIN diode
  A PIN diode is a diode with a wide, lightly doped 'near' intrinsic semiconductor region between a p-type semiconductor and an n-type semiconductor region. The p-type and n-type regions are typically heavily doped because they are used for ohmic contacts....
  
  , n-GaAs, and Si p⁺-n⁺-p-n^-, and ZnS:Mn structures.

• Nonlinear optical systems
  Nonlinear optics
  Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...
  
  : 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
  Feedback
  Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...
  
   into the input system via single-mirror feedback or a feedback loop. DSs may arise as bright spots in a 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
  Saturable absorption
  Saturable absorption is a property of materials where the absorption of light decreases with increasing light intensity. Most materials show some saturable absorption, but often only at very high optical intensities ....
  
  , degenerate optical parametric oscillator
  Optical parametric oscillator
  An optical parametric oscillator is a parametric oscillator which oscillates at optical frequencies. It converts an input laser wave into two output waves of lower frequency by means of second order nonlinear optical interaction. The sum of the output waves frequencies is equal to the input wave...
  
  s (DOPOs), liquid crystal light valve
  Light valve
  A light valve is a device for varying the quantity of light, from a source, which reaches a target. Examples of targets are computer screen surfaces, or a wall screen in the case of a light projector....
  
  s (LCLVs), alkali vapor systems, photorefractive media
  Photorefractive effect
  The photorefractive effect is a nonlinear optical effect seen in certain crystals and other materials that respond to light by altering their refractive index....
  
  , and semiconductor microresonators.

• If the vectorial properties of DSs are considered,vector dissipative soliton could also be observed in a fiber laser passively mode locked through saturable absorber,

• In addition,multiwavelength dissipative soliton in an all normal dispersion fiber laser passively mode-locked with a SESAM has been obtained. It is confirmed that depending on the cavity birefringence, stable single-, dual- and triple-wavelength dissipative soliton can be formed in the laser. Its generation mechanism can be traced back to the nature of dissipative soliton.

• Chemical systems: realized either as one- and two-dimensional reactors or via catalytic surfaces, DSs appear as pulses (often as propagating pulses) of increased concentration or temperature. Typical reactions are the Belousov-Zhabotinsky reaction
  Belousov-Zhabotinsky reaction
  A Belousov–Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in these oscillating systems is the inclusion of bromine...
  
  , the ferrocyanide-iodate-sulphite reaction as well as the oxidation of hydrogen, CO, or iron. Nerve pulses
  Action potential
  In physiology, an action potential is a short-lasting event in which the electrical membrane potential of a cell rapidly rises and falls, following a consistent trajectory. Action potentials occur in several types of animal cells, called excitable cells, which include neurons, muscle cells, and...
  
   or migraine aura waves also belong to this class of systems.

• Vibrated media: vertically shaken granular media, colloidal suspensions
  Colloid
  A colloid is a substance microscopically dispersed evenly throughout another substance.A colloidal system consists of two separate phases: a dispersed phase and a continuous phase . A colloidal system may be solid, liquid, or gaseous.Many familiar substances are colloids, as shown in the chart below...
  
  , and Newtonian fluid
  Newtonian fluid
  A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...
  
  s produce harmonically or sub-harmonically oscillating heaps of material, which are usually called oscillon
  Oscillon
  In physics, an oscillon is a soliton-like phenomenon that occurs in granular and other dissipative media. Oscillons in granular media result from vertically vibrating a plate with a layer of uniform particles placed freely on top...
  
  s.

• Hydrodynamic systems: the most prominent realization of DSs are domains of convection
  Horizontal convective rolls
  Horizontal convective rolls, also known as horizontal roll vortices or cloud streets, are long rolls of counter-rotating air that are oriented approximately parallel to the ground in the planetary boundary layer...
  
   rolls on a conducting background state in binary liquids. Another example is a film dragging in a rotating cylindric pipe filled with oil.

• Electrical networks: large one- or two-dimensional arrays of coupled cells with a nonlinear current-voltage characteristic
  Current-voltage characteristic
  A current–voltage characteristic is a relationship, typically represented as a chart or graph, between an electric current and a corresponding voltage, or potential difference.-In electronics:...
  
  . 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 are similar in spite of the microscopic differences. Typical observations are (intrinsic) propagation, scattering

Scattering

Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...

, formation of bound state

Bound state

In physics, a bound state describes a system where a particle is subject to a potential such that the particle has a tendency to remain localised in one or more regions of space...

s 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. Discrete difference equations and
cellular automata

Cellular automaton

A cellular automaton is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off"...

 are also used. Up to now,
modeling from first principles followed by a quantitative
comparison of experiment and theory has been performed only
rarely and sometimes also poses severe problems because of 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 system
  Reaction–diffusion system
  Reaction–diffusion systems are mathematical models which explain how the concentration of one or more substances distributed in space changes under the influence of two processes: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the...
  
  s, used for chemical systems, gas-discharges and semiconductors. The evolution of the state vector q(x, t) describing the concentration of the different reagents is determined by diffusion as well as local reactions:

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

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. Among other effects, one finds periodic oscillations of DSs ("breathing"), bound states, and collisions, merging, generation and annihilation.

• Ginzburg-Landau type systems
  Ginzburg-Landau theory
  In physics, Ginzburg–Landau theory, named after Vitaly Lazarevich Ginzburg and Lev Landau, is a mathematical theory used to model superconductivity. It does not purport to explain the microscopic mechanisms giving rise to superconductivity...
  
   for a complex scalar q(x, t) used to describe nonlinear optical systems, plasmas, Bose-Einstein condensation, liquid crystals and granular media. A frequently found example is the cubic-quintic subcritical Ginzburg-Landau equation

To understand the mechanisms leading to the formation of DSs, one may consider the energy ρ = |q|² for which one may derive the continuity equation

One can thereby show that energy is generally 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, propagating clusters in 2d, bound states and vortex solitons, as well as "exploding DSs".

• The Swift-Hohenberg equation
  Swift-Hohenberg equation
  The Swift–Hohenberg equation is a partial differential equation noted for its pattern-forming behaviour...
  
   is used in nonlinear optics and in the granular media dynamics of flames or electroconvection. Swift-Hohenberg can be considered as an extension of the Ginzburg-Landau equation. It can be written as

For d_r > 0 one essentially has the same mechanisms as in the Ginzburg-Landau equation. For 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. This also holds for the complex Swift-Hohenberg equations; however, propagating DSs as well as interaction phenomena are also possible, and observations include merging and interpenetration.

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.

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


for the velocity v of the DS, here σ
represents the bifurcation parameter and σ₀
the bifurcation point. For a bifurcation to a "breathing" DS,
one finds the Hopf normal form


for the amplitude A of the oscillation. name="gurevich"/> It is also possible to treat "weak interaction"
as long as the overlap of the DSs is not too large. In this way, a
comparison between experiment and theory is facilitated.,

Note that the above problems do not arise for classical
solitons as inverse scattering theory yields complete
analytical solutions.

See also

• soliton
  Soliton
  In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
  
  
• vector soliton
  Vector soliton
  In physical optics or wave optics, a vector soliton is a solitary wave with multiple components coupled together that maintains its shape during propagation. Ordinary solitons maintain their shape but have effectively only one polarization component, while vector solitons have two distinct...
  
  
• fiber laser
  Fiber laser
  A fiber laser or fibre laser is a laser in which the active gain medium is an optical fiber doped with rare-earth elements such as erbium, ytterbium, neodymium, dysprosium, praseodymium, and thulium. They are related to doped fiber amplifiers, which provide light amplification without lasing...
  
  
• Nonlinear system
• compacton, a soliton with compact support
• Clapotis
  Clapotis
  In hydrodynamics, the clapotis is a non-breaking standing wave pattern, caused for example, by the reflection of a traveling surface wave train from a near vertical shoreline like a breakwater, seawall or steep cliff....
  
  
• Freak wave
  Freak wave
  Rogue waves are relatively large and spontaneous ocean surface waves that occur far out in sea, and are a threat even to large ships and ocean liners...
  
  s may be a related phenomenon.
• Oscillon
  Oscillon
  In physics, an oscillon is a soliton-like phenomenon that occurs in granular and other dissipative media. Oscillons in granular media result from vertically vibrating a plate with a layer of uniform particles placed freely on top...
  
  s
• peakon, a soliton with a non-differentiable peak.
• Q-ball
  Q-Ball
  In theoretical physics, Q-ball refers to a type of non-topological soliton. A soliton is a localized field configuration that is stable—it cannot spread out and dissipate. In the case of a non-topological soliton, the stability is guaranteed by a conserved charge: the soliton has lower energy per...
  
   a non-topological soliton
• Soliton (topological).
• Soliton (optics)
  Soliton (optics)
  In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons:...
  
  
• Soliton model
  Soliton model
  The Soliton model in neuroscience is a recently developed model that attempts to explain how signals are conducted within neurons. It proposes that the signals travel along the cell's membrane in the form of certain kinds of sound pulses known as solitons...
  
   of nerve impulse propagation
• Spatial soliton
• Solitary wave
  Solitary wave
  In mathematics and physics, a solitary wave can refer to* The solitary wave or wave of translation, as observed by John Scott Russell in the Union Canal, near Edinburgh in 1834...
  
  s in discrete media http://www.livescience.com/technology/050614_baby_waves.html
• Topological quantum number
  Topological quantum number
  In physics, a topological quantum number is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations...
  
  
• Sine-Gordon equation
  Sine-Gordon equation
  The sine–Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally considered in the nineteenth century in the course of study of surfaces of constant negative...
  
  
• graphene
  Graphene
  Graphene is an allotrope of carbon, whose structure is one-atom-thick planar sheets of sp2-bonded carbon atoms that are densely packed in a honeycomb crystal lattice. The term graphene was coined as a combination of graphite and the suffix -ene by Hanns-Peter Boehm, who described single-layer...