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. 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 (conservative) 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 (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 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 (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 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 	Dissipative soliton
25050583	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 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 σ 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.