Linear and Nonlinear Wave Concepts
Linear and nonlinear wave phenomena occur in the propagation of waves through transparent media as well as in the reflection and transmission of such waves by opaque media. In the nonlinear case there are usually wave dependent changes in the material media supporting the waves and one requires self-consistent field descriptions both of the wave and material systems. Such fields possess rapidly varying; space-time components coupled to slowly varying components. In case of stochastic fields the former are identified as fluctuating and the latter as background (average) field components. This identification is not always precise, but when relevant, it introduces a certain degree of conceptual coherence in the treatment of excitation and response phenomena within both nonlinear and turbulent systems.
For rapidly varying wavepacket fields, detailed or microscopic features may not be of primary interest. More interesting in practice are the macroscopic averaged measures of the relatively slowly varying space-time dependent amplitude, frequency, wavenumber of such fields. In appropriate ranges, the average and "correlation" measures of rapidly varying deterministic or stochastic wavepackets can often be described either in terms of "rays" or in terms of the density, energy, and momentum dynamics of "quasiparticle" systems. The quasiparticle description of classical wavepackets, when valid, is analogous to the particle description of a quantum wave system in the limit h->0 wherein wave-particle duality concepts are valid in quantum mechanics.
Consider the problem of describing the overall response to excitation of a system containing a non-linear medium. At time t=0 a pulse or wavepacket in free space is incident on a semi-infinite planar non-linear medium. For sufficiently low power levels, one observes at time t>0 that reflected and transmitted pulses create no changes in the background medium. On the other hand, at sufficiently large power levels, changes in the background medium occur and there appears a rich nonlinear variety of wavepackets indicative of different " mode type" responses in the system. Many response features such as nonlinear chaos and strong turbulence may then develop and may not be completely understood. The following does not presume to present a complete overview of current understanding of non-linear phenomena but rather seeks to contrast wave descriptions in