The fields of linear waveguides that change `slowly', but otherwise arbitrarily in the direction of propagation, have an elegant description [18]. The modes are then locally those of the axially uniform waveguide. This adiabatic approximation applies when the wavelength of spatial change is long compared to relevant electromagnetic lengths. It can be borrowed directly for nonlinear waves.
For example, self-guided beams of a self-focusing material, with either
loss or gain, will self-taper. The beam power 
changes
according to the standard relation [18] of linear
waveguides, where 
, and e is a solution of (4).
When the linear part of the imaginary refractive index 
has loss or
gain, then 
is a constant and the power changes
exponentially with propagation distance z. For a spatial
soliton of a Kerr medium (Example 1, Section 4), the
characteristic soliton width 
is inversely proportional
to its power P. The soliton then tapers
exponentially with propagation distance `z' (see Ref. [8] for details).
When the nonlinear part of the
refractive index is imaginary as in two photon absorption, then
(for a Kerr material) 
is proportional to 
. This leads
to [8] 
, so that 
is now a linear function of propagation distance `z'.