From the linear perspective, harmonic generation arises when
light at frequency 
causes the molecules to vibrate
sinusoidally in time. Temporal propagation in such a medium is
thus analogous to spatial propagation through a periodic
medium.
Previous approaches[60]
[64] to
soliton propagation in a medium which exhibits harmonic
generation are based exclusively on the `new' physics and
mathematics of the phase-involved parametric interactions
rather than the familiar concept of refractive index change
discussed above. But, the elementary concepts of induced
linear waveguides apply elegantly to such parametric solitons,
leading to an intuitive description. In other words, both
soliton classes, for cubic and quadratic nonlinearities, can
be treated identically.
The familiar equations [16] for propagation in a medium
exhibiting harmonic generation can be cast into the linear
framework by simply identifying certain quantities as an
effective refractive index. Taking the fields to be linearly polarized
along the optical axes, this leads to 
, where c.c. is the complex conjugate, and where
and 
. Here 
and 
see two
different nonuniform waveguides 
and 
respectively with 
and
with 
denotes the linear part of the
refractive index and 
is the second order
susceptibility.
In general, the refractive index is complex which corresponds
to loss and gain. Only when both 
and
are real can stationary waves exist. This demands
that 
and 
are in phase or 
out-of-phase.
For plane wave propagation, 
and 
.
With the consistency relation that 
and
, then 
(where 
corresponds to
in or 
out-of-phase with each other).
Similarly for solitons, the envelopes 
and
`see' a linear waveguide characterized
by 
and 
, respectively. Thus, from Eq.
(
) the envelopes 
and 
satisfy
the modal equations
where the effective refractive indices 
and
are respectively
These equations are analogous to those for vector solitons (Sec. 3.1).
Thus, soliton propagation in a 
medium is like a
mode of a birefringent linear waveguide.
Suppose 
sees a sech
waveguide, i.e. 
whose mode
is 
with 
where s is given by 
.
Self-consistency ahows that 
and 
both
see a sech
waveguide with 
, and
where 
and 
are given by: 
so that
and 
. The soliton Eq.
() is the only solution known analytically
[60,62].
In the general case, no analytical form exists for
the solitons in 
medium, and the stationary
solutions of Eqs. (), () are known only
numerically [62]. Finally, cascading is a particular limit of the
stationary soliton, when the 
term in Eq. ()
can be neglected.
