The conventional approach to nonlinear waves is highly mathematical. It conveys the impression that nonlinear physics is separate from linear physics, being composed of radical new phenomena. While linear physics forms the foundations for our intuition, nonetheless it has appeared to be of little use for understanding nonlinear waves.
In sharp contrast, we have demonstrated that linear physics provides a natural approach to nonlinear guided waves. It imparts deep physical insight which in turn, generates the fundamental equations, allows for unforeseen generalizations and motivates important discoveries. These include: dynamic solitons, stability criteria, optical vortex solitons, spiralling solitons and a physical explanation for why certain self-guided beams have radiation free collisons as well as why higher order solitons result from the process of scaling up the fundamental soliton.
Although it has long been realized [2] that a self guided beam is a mode of the axially uniform optical waveguide it induces; nonetheless, this elementary concept has received virtually no attention over the years, apart from being used passively [65]. Ours would appear to be the first paper to recognize that the linear perspective can be generalized into a rigorous and systematic approach to nonlinear guided waves, even in a medium that exhibits second harmonic generation.
This new conceptual approach is demonstrated here for spatial waves, but it can also be developed for temporal pulses. However, the linear equivalents of spatial waves concern well understood optical waveguides, whereas those for pulses are rather more abstract [66]. Accordingly, it is easier to conceptualize in the spatial domain and then to translate novel findings into the temporal domain.
Finally, the linear perspective facilitates a qualitative
theory of nonlinear waves, whereby the salient characteristics
and stability are `read' from the graph of the material
nonlinearity, 
. We demonstrate this in the appendix.
Clearly all approaches to nonlinear waves confront the same equations, but the linear perspective guides us in a physical manner to these equations as well as to their classes of solutions. It emphasizes that linear and nonlinear guided waves share common concepts and that they both are described by the elegant mathematical methods already existing in the literature of linear optical waveguides. In this way, beginners can quickly acquire expertise.
Not everyone likes physical arguments, but they can offer great potential for creativity. It is in the best scientific tradition to build first from elementary physics and to exhaust all of its possibilities. Indeed, we are optimistic that the linear perspective has potential to further expand our understanding of nonlinear science.
Acknowledgements
One of us (AWS) was invited to present an overview of this research at the 1995 Gordon Conference on Nonlinear Optics and Lasers chaired by Y. Silberberg. We appreciate the invitation to publish this work in the forthcoming book ``Research Trends in Physics: Nonlinear and Quantum Optics, N. Bloembergen (Editor-in-Chief)''. It appears by invitation in MPLB instead. The authors are members of the Australian Photonics Cooperative Research Centre.
References