Soliton Systems in the Presence of Amplification, Irregularities, and Damping

Abstract

In the past, nonlinear physics received tremendous attention mainly for two reasons. First, in the late sixties the soliton was detected as a robust mode following from integrability of some nonlinear partial differential equations (PDEs)1. And in many brilliant papers it was shown that the soliton concept is not restricted to a very narrow area of physics; on the contrary, it has many and broad applications in physics and biology. The second milestone of nonlinear physics was the detection of chaos2 in simple deterministic systems modelled by mappings or nonlinear ordinary differential equations (ODEs). For a long time both directions were investigated in parallel with not too many intersections. The only small overlap is understandable: soliton physics deals with complicated integrable systems while chaos investigations elucidate the aspect of non-integrability in simple nonlinear models. A link between both directions is the solitary wave. The latter is a localized solution of soliton form, but it does not require integrability of the underlying model. Then, e. g., we cannot expect form invariance during nonlinear interactions, as it is a fascinating outcome of the integrable models through the inverse scattering transform (IST). The perturbations may be considered in the structural sense, and from the point of soliton physics we are faced with both problems, Lyapunov and structural stability. In the near-integrable cases perturbed soliton solutions can exhibit some very interesting and generic nonlinear dynamics. Transition to chaos in time and/or space may occur. Solitary wave solutions or localized spatially coherent states can also exist far from integrability. In that sense one should consider them on the same footing as the more general stationary nonlinear solutions investigated during the past decades in, e. g., hydrodynamics and plasma physics.KeywordsSolitary WaveLocalize ModeSoliton SolutionSolitary Wave SolutionAnderson LocalizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.