With a topic as general as Wave Propagation in Nonlinear Dispersive Xedia, there are two approaches that come to mind with regard to writing a short essay on the subject. The traditional approach is historical, having the advantage of exposing readers to discoveries as they happen in time, so that the reason for a particular approach is clearly recognized, but at the expense of length and statements like "This was found much later...". The unorthodox approach is based on seeing things at this instant of time, realizing what is important and worthy of inclusion without paying much heed to when they were placed in the time capsule. Working under the constraints of a limited length, the author is therefore forced to depart from the traditional values and would, in advance, like to express sincere apologies to contributors in this area who had to be omitted.

Perhaps an appropriate introduction to the subject of wave propagation in general would be to examine what exactly constitutes a save. To quote Whitham [l], "various restrictive definitions can be given, but to cover the whole range of wave phenomena it seems preferable to be guided by the intuitive view that a wave is any recognizable signal that is transferred from one part of the medium to another with a recognizable velocity of propagation. The signal may be any feature of the disturbance, such as a maximum or an abrupt change in some quantity, provided that it can be clearly recognized and its location at any time can be determined. The signal can distort, change its magnitude, and change its velocity, provided it is still recognizable. This may seem a little vague, but it turns out to be perfectly adequate and any attempt to be more precise appears to be too restrictive; different features are important in different types of waves."

The organization of this article may now be summarized as follows: Section II exposes the rudiments of non-linearity and dispersion. Sections III and IV describe some effects of nonlinear nondispersive/dispersive propogation for CW waves, viz., harmonic (and sub harmonic) generation and self-refraction respectively. Sections V and VI outline the principles behind a possible balance between non-linearity and dispersion for baseband and envelope pulsed propagation leading to solitary wave/soliton formation. Finally, Section VII briefly summarizes some of the possible applications of soliton theory in physics and engineering.