Dissipative solitons (DSs) are stable solitary localized
structures that arise in nonlinear spatially extended
dissipative systems due to mechanisms 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.
DSs have been experimentally observed for a long time.
Helmholtz^{[1]} measured the propagation velocity of nerve pulses in
1850. In 1902, Lehmann^{[2]} 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 in 1834.^{[3]}
These observations initiated the theoretical work of
Rayleigh^{[4]} and Boussinesq^{[5]} 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.^{[6]}
On this background the term "soliton" was
coined by Zabusky and Kruskal^{[7]} 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.^{[8]} introduced the inverse scattering technique
for solving the KdV equation and proved that this equation is
completely integrable. In 1972 Zakharov and
Shabat^{[9]} 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
Sine-Gordon 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 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 small energy losses into account as a
perturbation, and on a long scale the amplitude of the soliton
will decay and finally vanish.^{[10]}
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^{[11]} in 1952), since
1990 the amount of research has significantly increased (see e.g.
^{[12]}^{[13]})
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: 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 a high-ohmic barrier, ^{[14]} AC systems with a dielectric barrier,^{[15]} and as anode spots,^{[16]} as well as in an obstructed discharge with metallic electrodes.^{[17]}
DSs experimentally observed in planar dc gas-discharge systems with high-ohmic barrier
Semiconductor 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,^{[18]} n-GaAs,^{[19]} and Si p^{+}-n^{+}-p-n^{-},^{[20]} and ZnS:Mn structures.^{[21]}
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 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,^{[22]} degenerate optical parametric oscillators (DOPOs),^{[23]} liquid crystal light valves (LCLVs),^{[24]} alkali vapor systems,^{[25]} photorefractive media,^{[26]} and semiconductor microresonators.^{[27]}
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,^{[28]} the ferrocyanide-iodate-sulphite reaction as well as the oxidation of hydrogen,^{[29]} CO,^{[30]} or iron.^{[31]}Nerve pulses^{[32]} also belong to this class of systems.
Vibrated media: vertically shaken granular media,^{[33]}colloidal suspensions,^{[34]} and Newtonian fluids^{[35]} 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.^{[36]} Another example is a film dragging in a rotating cylindric pipe filled with oil.^{[37]}
Electrical networks: large one- or two-dimensional arrays of coupled cells with a nonlinear current-voltage characteristic.^{[38]} 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, 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. Discrete difference equations and
cellular automata 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 systems, used for chemical systems, gas-discharges and semiconductors.^{[39]} 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.^{[40]} Among other effects, one finds periodic oscillations of DSs ("breathing"),^{[41]}^{[42]} bound states,^{[43]} and collisions, merging, generation and annihilation.^{[44]}
Ginzburg-Landau type systems for a complex scalar q(x, t) used to describe nonlinear optical systems, plasmas, Bose-Einstein condensation, liquid crystals and granular media.^{[45]} A frequently found example is the cubic-quintic subcritical Ginzburg-Landau equation
An alternative formulation is given by considering the energy ρ = |q|^{2} for which one may derive the continuity equation
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,^{[46]} propagating clusters in 2d,^{[47]} bound states and vortex solitons,^{[48]} as well as "exploding DSs".^{[49]}
The Swift-Hohenberg equation 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.^{[50]} 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.^{[51]} 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.^{[52]}
Space-time plots showing the dynamics and interaction of DSs as numerical solutions of the above model equations in one spatial dimension
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.^{[53]}
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,^{[54]} here σ
represents the bifurcation parameter and σ_{0}
the bifurcation point. For a bifurcation to a "breathing" DS,
one finds the Hopf normal form
for the amplitude A of the oscillation.^{[42]} In this way, a
comparison between experiment and theory is facilitated.^{[55]},
^{[56]}
Note that the above problems do not arise for classical
solitons as inverse scattering theory yields complete
analytical solutions.
References
^ H. Helmholtz, Arch. Anat. Physiol. 57 (1850):
276
^ C. P. Schenk et al., Phys. Rev. Lett. 78 (1997): 3781
^ I. S. Aranson and L. Kramer, Rev. Mod. Phys. 74 (2002): 99
^ V. V. Afanasjev et al., Phys. Rev. E 53 (1996): 1931
^ N. N. Rosanov et al., J. Exp. Theor. Phys. 102 (2006): 547
^ L.-C. Crasovan et al., Phys. Rev. E 63 (2000): 016605
^ J. M. Soto-Crespo et al., Phys. Rev. Lett. 85 (2000), 2937
^ J. M.
Soto-Crespo and N. Akhmediev, Phys. Rev. E 66 (2002): 066610
^ H. Sakaguchi and H. R. Brandt,Physica D 97 (1996): 274
^ H. Sakaguchi and H. R. Brandt, Physica D 117 (1998): 95
^ R. Friedrich,
Group Theoretic Methods in the Theory of Pattern Formation, in
Collective dynamics of nonlinear and disordered systems,
Springer (2004)