Self guided beams of a homogeneous medium are known to attract or
repel when they are like polarized. But, the elementary physics of
parallel linear waveguides plus self consistency suggests that they
remain parallel if the even and odd modes of their induced two linear
waveguide system are orthogonally polarized [33]. Thus, the
composite field comprising the soliton pair has the form of (2)
with subscripts `a' and `b' denoting respectively the even (+) and
odd 
modes. The usual perturbation theory [30]
for finding modes of parallel linear waveguides can then be
used to describe the polarization dynamics of parallel well
separated solitons. For a nonlinear material which exhibits
induced isotropy 
, this shows that the two distant
orthogonally polarized solitons can be arbitrarily polarized,
whereas they must be linearly polarized in the general
(
) Kerr medium.
For arbitrary soliton separation, (6) must be solved with 
,
respectively, the fundamental (+) even mode and the first 
odd
mode. This pair of coupled equations has a simple analytical solution in
one dimension 
for an isotropic Kerr medium
[33]. For a birefringent Kerr material, the even and odd modes
must be circularly polarized with 
, 
now
, 
as discussed in the above example. Numerical studies of this
equation, in
a different physical context, proves existence [31]. For any
given soliton
power, there is a soliton separation for which 
. The
dynamic soliton is then a vector (stationary) soliton in this
medium whose birefringence is induced (see Section 3.1).