Dissipative soliton

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.Averaged current density distribution without oscillatory tails.Averaged current density distribution with oscillatory tails.Single 'breathing' DS as solution of the two-component reaction-diffusion system with activator u (left half) and inhibitor v (right half). 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 interesting behavior – e.g. scattering, creation and annihilation – all without the constraints of energy or momentumconservation. 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 measured the propagation velocity of nerve pulses in1850. In 1902, Lehmann found the formation of localized anodespots in long gas-discharge tubes. Nevertheless, the term'soliton' was originally developed in a different context. Thestarting point was the experimental detection of 'solitarywater waves' by Russell in 1834.These observations initiated the theoretical work ofRayleigh and Boussinesq around1870, which finally led to the approximate description of suchwaves by Korteweg and de Vries in 1895; that description is known today as the (conservative)KdV equation. On this background the term 'soliton' wascoined by Zabusky and Kruskal in 1965. Theseauthors investigated certain well localised solitary solutionsof the KdV equation and named these objects solitons. Amongother things they demonstrated that in 1-dimensional spacesolitons exist, e.g. in the form of two unidirectionallypropagating pulses with different size and speed and exhibiting theremarkable property that number, shape and size are the samebefore and after collision. Gardner et al. introduced the inverse scattering techniquefor solving the KdV equation and proved that this equation iscompletely integrable. In 1972 Zakharov andShabat found another integrable equation andfinally it turned out that the inverse scattering technique canbe applied successfully to a whole class of equations (e.g. thenonlinear Schrödinger andsine-Gordon equations). From 1965up to about 1975, a common agreement was reached: to reserve the term soliton topulse-like solitary solutions of conservative nonlinear partialdifferential equations that can be solved by using the inversescattering technique. With increasing knowledge of classical solitons, possibletechnical applicability came into perspective, with the mostpromising one at present being the transmission of opticalsolitons via glass fibers for the purpose ofdata transmission. In contrast to conservative systems, solitons in fibers dissipate energy andthis cannot be neglected on an intermediate and long timescale. Nevertheless, the concept of a classical soliton canstill be used in the sense that on a short time scaledissipation of energy can be neglected. On an intermediate timescale one has to take small energy losses into account as aperturbation, and on a long scale the amplitude of the solitonwill decay and finally vanish. There are however various types of systems which are capable ofproducing solitary structures and in which dissipation plays anessential role for their formation and stabilization. Althoughresearch on certain types of these DSs has been carried out fora long time (for example, see the research on nerve pulses culminatingin the work of Hodgkin and Huxley in 1952), since1990 the amount of research has significantly increased (see e.g. )Possible reasons are improved experimental devices andanalytical techniques, as well as the availability of morepowerful computers for numerical computations. Nowadays, it iscommon to use the term dissipative solitons for solitary structures instrongly dissipative systems.