Dissipative soliton


Dissipative solitons [Inverse scattering transform|] 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 interesting behavior - e.g. scattering, creation 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.
Helmholtz measured the propagation velocity of nerve pulses in
1850. In 1902, Lehmann 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.
These observations initiated the theoretical work of
Rayleigh 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
KdV equation.
On this background the term "soliton" was
coined by Zabusky 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 et al. introduced the
inverse scattering technique
for solving the KdV equation and proved that this equation is
completely integrable. In 1972 Zakharov 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. 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 conservative systems, 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, 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
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 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
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. 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.

Inline

Books and overview articles