Abstract
The creation of stable 1D and 2D localized modes in lossy nonlinear media is a fundamental problem in optics and plasmonics. This article gives a mini review of theoretical methods elaborated on for this purpose, using localized gain applied at one or several hot spots (HS). The introduction surveys a broad class of models for which this approach was developed. Other sections focus in some detail on basic 1D continuous and discrete systems, where the results can be obtained, partly or fully, in an analytical form (and verified by comparison with numerical results), which provides deeper insight into the nonlinear dynamics of optical beams in dissipative nonlinear media. Considered, in particular, are the single and double HS in the usual waveguide with the self-focusing (SF) or self-defocusing (SDF) Kerr nonlinearity, which gives rise to rather sophisticated results in spite of apparent simplicity of the model, solitons attached to a PT-symmetric dipole embedded into the SF or SDF medium, gap solitons pinned to an HS in a Bragg grating, and discrete solitons in a 1D lattice with a hot site.
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