Of the myriad of possibilities, those beams that induce waveguides which are uniform in the direction of propagation comprise the simplest class. We investigate these first and then go on to the general class.
Linear waves: If a linear waveguide is axially
uniform with a refractive index profile
, then its electric field vector
can be represented as a sum of modes
where 
and 
, 
are the modal
eigenfunctions and propagation constants, respectively.
The vector field E is a solution to the homogeneous Maxwell's equations. However, waveguides of practical interest have maximum and minimum refractive index values that are nearly equal. It is then well known that [18]
which holds exactly for TE waves only, where 
and 
is the tensor refractive index. We discuss the nature of this
approximation and the nonlinear Schrödinger equation at length in
Section 8.
Nonlinear waves: We have been discussing the fields of
linear waveguides that are axially uniform. Now, if
Eq. (3) is to describe nonlinear beams, then the
self-consistency relation (1), demands that their induced
waveguides, 
is unchanged in the
direction of propagation. Thus, nonlinear beams can only be
represented by one mode, by two orthogonal modes obeying
or even by two like
polarized modes of different wavelength when four wave mixing
can be neglected [19].
Equation (3) specifies the complete class of beams whose induced waveguides are axial uniform. No other representation will induce an axially uniform waveguide. Despite its simplicity, Eq. (3) describes a multitude of nonlinear waves including classes of self-guided beams of a homogenous medium, surface waves at a nonlinear interface and modes of nonlinear waveguides.
We classify beams according to whether they are one or two modes of the induced waveguide. If the beam is one mode of its induced waveguide, its polarization is maintained with propagation. These are traditional `stationary' solutions, e.g. the classical sech soliton [2], and vector solitons [20]. If the beam is two modes of its induced waveguide, then novel polarization dynamics arise due to modal beating. These are the new `dynamic' solitons [21] whose members describe numerous physical situations [22,23].
Surprisingly, it has taken some thirty years to generalize the concept of an optical self guided beam, from that of one mode of the linear waveguide it induces [2], to that of two modes [21]. This is remarkable, because no additional mathematics is required to pass from the classical stationary waves to the more general `dynamic' waves, only physical insight. Apparently, the rationale for the dynamic waves is not obvious outside the framework of the linear perspective. Indeed, the linear modal expansion of (2) is heresy within the formal mathematics of solitons [24], whereas it is natural within the linear perspective of the spatial domain advanced here.
This completes the conceptual foundations of the linear perspective. Having motivated the general form for solitons, Eq. (2), it is straightforward to find solutions of the wave equation for any specified nonlinear medium. As examples, we give some discoveries that were motivated by the linear perspective, including vortex solitons, vector rotating solitons, parallel solitons and light guiding light. Readers who are primarily concerned with the framework of the linear perspective should pass directly to Section 4.
