The familiar sech
profile
linear waveguide has the form 
,
where 
, 
and 
is a characteristic half-width. Taking the modal
field amplitude 
, where 
, leads to
with 
. Here V is the waveguide
parameter 
, and the modal propagation constant
is, 
. The
sech
waveguide is single moded provided that 
or, equivalently, 
.
To find the nonlinearity that supports solitons that have a
sech
modal field, we use the self consistency
relation (Fig. 1) that 
, where
is the material nonlinearity. It is necessary
to invert 
to obtain an explicit relation for the
dependence of the refractive index on intensity, i.e. 
, where 
. This leads to a power-law nonlinearity of
the form 
with 
. To obtain a solution of the nonlinear problem, we
replace s by 
. For example, this specifies the
parameter V of the soliton-induced waveguide in terms of the
power of the nonlinearity, q, leading to 
. The
soliton intensity (
) is 
. Further details can be found in [32].