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Abstract Dissipative optical solitons (autosolitons) are considered, analytically and numerically, for a wide-aperture laser with saturable absorption in the case when nonlinearity relaxation and non-locality are important. Depending on the values of the relaxation times and diffusion coefficients for gain and absorption, we find stable stationary dissipative solitons which are similar to those in class A lasers, slowly moving dissipative solitons, and also a new type of “flying” soliton which is characterized by a high velocity of motion and a high peak intensity.

1 Introduction Dissipative solitons, or autosolitons, first discovered in mechanics in 1831 [1] were further intensively investigated in the physics of electric discharge and in biology, chemistry, and many other fields of science [2, 3]. Mathematically, the governing equations were mainly of reaction–diffusion type for the dynamics of the nonlinear interactions of some “inhibitors” and “activators” in space and time [2]. The turn of optics began with the invention of lasers whose high-power radiation provides high nonlinearity and where different types of pumping are able to compensate for various types of energy losses. In addition to features in common with other dissipative solitons, the optical forms have some new ones. For example, diffraction N.N. Rosanov Vavilov State Optical Institute, Institute for Laser Physics, Birzhevaya Liniya 12, Saint-Petersburg 199034, Russia, [email protected] S.V. Fedorov Vavilov State Optical Institute, Institute for Laser Physics, Birzhevaya Liniya 12, Saint-Petersburg 199034, Russia, [email protected] A.N. Shatsev Vavilov State Optical Institute, Institute for Laser Physics, Birzhevaya Liniya 12, Saint-Petersburg 199034, Russia, [email protected] Rosanov, N.N. et al.: Dissipative Solitons in Laser Systems with Non-local and Non-instantaneous Nonlinearity. Lect. Notes Phys. 751, 93–111 (2008) c Springer-Verlag Berlin Heidelberg 2008 DOI 10.1007/978-3-540-78217-9 4 

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is typical for optics and replaces the traditional diffusion. With the corresponding diffractive oscillations of the electric field, these dissipative solitons could be called “diffractive dissipative solitons” [4]. In a sense, nonlinear optics and laser physics provide generally simpler and more tractable examples of dissipative solitons, and at the moment many of their features are best known in the area of optics. Dissipative optical solitons, predicted first in the 1980s, are now objects of intensive research due to their rich and unusual physics and great potential for applications [3, 5, 6]. Due to the increase in the number of researchers and the diversity of laser systems, the actual definition of dissipative solitons varies in different publications. Here, to avoid misunderstanding, stable localized structures in homogeneous or weakly modulated nonlinear media or systems with essential energy exchange (energy sinks and sources) will be labeled as dissipative solitons or autosolitons [7]. The possibility of moderately