Nonlinear science is believed by many to be the most deeply important frontier for understanding nature [1]. However, it is often promoted as being quite separate from linear science, embracing radical new phenomena such as solitons, multistabilities, bifurcations, etc. Indeed, the President of the American Physical Society states [1] ``In my research over the past two decades I have been fascinated by a set of developments in ``nonlinear science". A particularly remarkable manifestation is the entirely counterintuitive excitation called the soliton ...". So, nonlinear science has a mystique because it appears to be divorced from our intuition, an intuition that has been built upon linear phenomena.
We wish to reverse this notion. Nonlinear phenomena can be approached from a linear perspective. This is not a new technique for solving nonlinear equations. Rather, the linear perspective;
(i) Motivates the possible classes of nonlinear waves and their characteristics. It thus predicts novel phenomena, such as solitons with internal dynamics, vortex solitons, spiralling solitons, self-tapering, light guiding light.
(ii) Facilitates unforeseen generalizations, such as those necessary for a universal criterion for stability.
(iii) Offers a powerful predictive tool, for example one which foreshadows rather mysterious properties of soliton collisions and higher order solitons.
(iv) Shows how the mathematical foundation for nonlinear waves is borrowed in an elegant form from the literature of linear optical waves.
(v) Allows for closed form solutions of illustrative examples to be lifted directly from the pages of linear physics.
This powerful approach is demonstrated here through the vehicle of
spatial guided wave optics, embracing such phenomena as guiding and
manipulating light by light itself
[2]
[14].
We begin in Section 2 with the notion of linear equivalence. Section 3 shows how this generalizes the concept of self guidance. In Section 4 solitons are found by `inverting' linear waves. The linear perspective is used in Sections 5 and 6 respectively to motivate the variational approximation and to provide a universal theory of stability. Section 7 provides physical insight for higher order solitons and for radiation free collisions. Section 8 presents the weakly guiding theory of nonlinear waves. Section 9 discusses solitons in a medium exhibiting second harmonic generation. A qualitative approach to solitons is presented in the appendix.