We now consider colliding self guiding beams of a homogeneous
isotropic medium. These beams induce a complicated criss-crossed (linear)
waveguide structure. Light travelling along one waveguide will
in general leak due to reflection and
scattering. From this physical perspective, it is surprising that the
waveguide parameters can ever be contrived to result in exactly radiation
free transmission. But, we know from the results of inverse scattering
[48] that the isotropic `X'-junction (linear) waveguide
induced by colliding one-dimensional self guided beams in an isotropic
Kerr medium is precisely radiation free. What is the physics
that underpins this important result?
To answer this fundamental question, we first recall that the radiation free nature of soliton collisions is independent of the relative size and also the relative polarization of the beams [48]. Thus, for convenience, we take one beam to be spread out to approximate a plane wave, while the other is of arbitrary size and polarized orthogonal to the first beam. Under these conditions, the quasi-plane wave does not affect the induced isotropic waveguide to first order and so sees the (linear) waveguide induced by the other beam.
Now, the plane wave will refract through the waveguide and, in general, it
will also be reflected. This is the elementary physics underlying the
phase shift and radiation associated with self-guided beams, in general.
However, we explained in example 2 of Section 4 that the sech refractive
profile waveguide is special in that it is reflectionless to all angles
of plane wave incidence, provided its characteristic waveguide parameter
, where V is defined in example 1 of Section 4. But, this
is precisely the waveguide induced by a one-dimensional self guided beam
propagating in an isotropic 
Kerr medium [28].
We have provided a physical explanation for the phase shift as well as
the peculiar radiation free nature of soliton collisions. Can we use
this elementary physics for predictions? Suppose, for example, that the
above Kerr medium solitons differ in their wavelength. To examine the
physical consequences of this, we again recognize that the (linear)
waveguide induced by the plane wave is exactly that seen by weak
(`signal') beam propagating along a waveguide induced by the (`pump')
soliton. This is discussed fully in example 4 of Section 3 for the case
when the signal and pump differ in wavelength. We showed that the signal
does not `see' a 
waveguide. Accordingly, the soliton
induced waveguide is no longer reflectionless to the incident plane
wave. This prediction of radiation arising from colliding solitons of
different wavelength is also consistent with recent numerical findings
[55].
Suppose next that the nonlinear material exhibits induced
birefringence as is the general case for a Kerr 
medium
[28]. How does this influence the collisions of orthogonally
polarized solitons, when the polarization is either linear or circular?
The above linear physics, together with the findings of example 4, Section
3, again predicts radiation.
Finally, linear physics also shows that if radiation is present during collisions, however small, then self guided beams can fuse or give birth to multiple solitons [14].