The simplest beam is one mode of the isotropic
(linear) waveguide it induces. From Eq. (2), we observe that
the beam polarization is arbitrary and independent of spatial
position. Then 
,
with 
a unit vector and e, 
found from
the eigenvalue equation
where 
is the transverse Laplacian.
In general, 
also depends explicitly on x, y. Solutions include
the `classical' bright [2] and dark [25,26]
one-dimensional
solitons of a cubic (Kerr) homogeneous nonlinear medium.
When the beam is one mode of the birefringent
(linear) waveguide it induces, then the polarization as well
as its dependence on positions x, y are set by the
characteristics of the birefringence. In general, the direction
of the optical axis of 
depends on spatial position
x, y as is known from linear waveguides [27].
However, the field can be expressed as 
when the directions of the optical
axis 
are independent of
position. From Eq. (3), this leads to
The simplest class of beams are polarized along one optical axis `a' or `b', e.g. those that are linearly or circularly polarized in a birefringent Kerr medium [28]. These induce an effectively isotropic waveguide.
Beams requiring both `a' and `b' components are called vector solitons [20]. The simplest conceptual example is for a homogeneous isotropic medium with induced birefringence [29], although examples including linear birefringence are usually discussed.
