In general, beams induce linear waveguides that are not axially uniform. These beams also obey the self-consistency relation discussed in Section 2 and the linear perspective leads us again to the wave equation (3). Unfortunately, there are few exact analytical solutions to this equation or to its Schrödinger type approximation (discussed below). The difficulty is innate to the subject and not to the method of approach.
Nevertheless, the linear perspective provides physical insight and approximate analytical forms for a number of important situations as shown below. In doing so, it also answers questions about seemingly mysterious phenomena, such as: Why does scaling up a fundamental soliton eventually result in it becoming a soliton that changes periodically with propagation (a second order soliton) ? Why do certain self guided beams have radiation free collisions ? Unlike the inverse scattering technique [48], our approach is not restricted to one-dimensional beams of an isotropic Kerr (cubic) nonlinear medium, so the equations need not be integrable.
We next give examples of beams that: collide, have periodic changes in intensity (including higher order solitons); attract, repel or spiral around each other; self-contract or expand; and exchange their power on nonlinear couplers.