Dear Colleagues,

Nonlinearity and dissipation (realized as a combination of losses and compensating gain) are fundamental properties featured by a vast class of physical settings. Most diverse and, arguably, most important examples of such settings occur in photonics, in the form of laser cavities, propagation of powerful laser beams in various materials, semiconductor microcavities filled by exciton-polariton condensates, etc. Other important examples of nonlinear dissipative systems occur in plasma physics, hydrodynamics (in particular, in thermal convection), chemical waves, etc. The nonlinearity is a crucially important ingredient of these settings when amplitudes of propagating waves and/or stationary patterns are sufficiently large.

Both linear and nonlinear properties of the underlying media are naturally classified as conservative and dissipative ones. At the linear level, the conservative properties represent temporal dispersion and/or spatial diffraction, as well as possible waveguiding structures, or effective trapping potentials induced, in particular, by spatial patterning of the refractive index in optics. Linear dissipative effects include losses and gain, as well as dispersion of the gain, i.e., diffusion or viscosity of the medium. The presence or absence of the latter effect is determined by the physical nature of the respective settings. For instance, in laser cavities diffusion is absent, as photons are not subject to diffusion. On the other hand, the diffusion of reactants plays a crucially important role in the propagation of chemical waves. At the nonlinear level, cubic conservative effects most typically represent self-focusing or self-defocusing of electromagnetic fields or matter waves, while dissipative cubic terms account for nonlinear losses, such as two-photon absorption in optics. Higher-order quintic conservative self-defocusing terms play a critically important role in two- and three-dimensional (2D and 3D) geometries, preventing the collapse which is driven by the usual cubic self-focusing. In many situations, such as in optical amplifiers integrated with saturable absorbers, the linear dissipative terms represent pure losses, while the gain is provided by the cubic term. To prevent the system from the blowup, it is then necessary to include a stabilizing element in the form of quintic losses. Another possibility to create stable patterns is to use nonlinearly-saturable absorption. On the other hand, if the cubic gain is applied locally, in a spatially localized region, the system can generate stable nonlinear patterns even in the absence of higher-order stabilizing losses. In some cases nonlinear effects in various media may be nonlocal, a well-known example being thermal nonlinearity in optics.

Stable states which may exist in nonlinear dissipative media must simultaneously satisfy two conditions: on the one hand, similar to conservative media, linear diffraction and/or dispersion must be brought in balance with nonlinear self-focusing, if one wants to create solitons, or with nonlinear self-defocusing in the case of delocalized states, such as dark solitons or domain walls in two-component systems. On the other hand, stable balance must also be secured between the gain and loss in active systems, or between the pump and loss in passive ones. In the generic case, the necessity to simultaneously satisfy two balance conditions does not admit existence of continuous families of states, unlike conservative settings, where the existence of continuous families (in particular, of solitons, parameterized by the intrinsic frequency) is a generic situation. Instead, nonlinear dissipative systems give rise to isolated states, which are often called dissipative solitons. If they are stable, they play the role of attractors in the respective model. A class of systems intermediate between the generic dissipative ones and conservative systems feature PT symmetry, i.e., symmetrically placed  gain and loss elements. In the presence of the nonlinearity, PT-symmetric systems, although being, in principle, dissipative ones, support continuous families of nonlinear states (in particular, PT-symmetric solitons), similar to conservative systems.

As concerns theoretical models of nonlinear dissipative media, many of them originate from the nonlinear Schroedinger equation (NLSE) with the cubic or more complex (e.g., cubic-quintic) nonlinearity. By itself, NLSE is a conservative model. Adding to it linear and nonlinear terms accounting for the gain and loss, one arrives at the class of complex Ginzburg-Landau equations (CGLEs) - roughly speaking, these are NLSEs with complex coefficients. In many cases, CGLEs give rise to localized solutions which represent dissipative solitons. Furthermore, in the 2D and 3D cases, one can readily find dissipative solitons with embedded vorticity. On the other hand, passive optical settings, pumped by external beams, are generically modeled by equations of the Lugiato-Lefever type, i.e., the NLSE with linear loss and an external pumping term.

The present Special Issue aims to collect original contributions, both theoretical and experimental ones, and reviews dealing with various topics belonging to the broad area of nonlinear dissipative systems. In particular, works addressing dissipative solitons (one- and multidimensional) and dissipative solitary vortices will be very appropriate for the publication in this Special Issue.

Prof. Boris Malomed
Guest Editor

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