It is often useful to have a simple approximation for waves whose intensity is axially uniform. Because these waves are the modes of the linear waveguide they induce, we should be able to borrow the elegant methods of linear waveguides without developing new approximations for the nonlinear wave equation.
For example, the familiar variational approximation of linear physics, when taken together with the self-consistency condition (1), gives the same results as the more complicated Lagrangian or Hamiltonian (nonlinear) approach [42]. This is another example of how mathematics can disguise the underlying linear nature of problems.
Recall the variational procedure for linear waveguides [18,43].
Taking
the modal field amplitude to be 
with
and 
the waveguide profile,
then 
has the familiar stationary representation
for isotropic material, where 
is expressed as a real function.
We next use this to find the fundamental nonlinear wave for a given
material nonlinearity 
. First, choose a trial function, say
, whose amplitude `a' and scaling parameter `
' are
unspecified. This trial function induces a linear waveguide
characterized by 
. Next, apply the standard variational
procedure
for finding 
for this profile.
Because the variational procedure applies to a fixed profile, the
dependence of 
in the profile n is ignored. We do this by labelling
it 
, so that the induced profile is 
. This leads to an expression for 
. Finally, self-consistency demands that the trial
function equal the nonlinear wave, requiring 
. This
last
equation interrelates 
with the modal amplitude. A number of examples
are available using the Gaussian trial function [44].
