Nonlinear beams are a theoretical curiosity unless they are stable to arbitrary perturbations of the initial conditions. Such perturbations potentially cause a significant change to the induced waveguide over a finite distance. The linear perspective is pivotal in the development of the universal stability criteria [45], because seemingly disparate nonlinear beams are shown to be topologically equivalent. Once the stability is known for one member of the class, it is known for all members.
For example, fundamental nonlinear beams (those without nodes) all induce a linear waveguide whose shape depends on the wave type. In this way, self guided beams, waves at a nonlinear interface and nonlinear modes of waveguides and couplers are all the same animal. Knowing the theory of stability for one type, gives the mathematical framework for finding the universal stability criterion for the entire class. Hence, the theory of stability for self guided beams [46] is easily extended to a universal criterion of stability reported in [45].
Stability is determined by simply reading
the familiar dispersion curves, modal propagation constant 
vs
modal power 
, where 
and e are found
from (4) and 
is the infinite cross section. These curves
require knowing only the stationary characteristics of beams as dictated
by (4) or equivalently, by the characteristics of the mode of the induced
(linear) waveguide. No further calculation is required. The
elementary rules for reading these curves is discussed
elsewhere [45].
Linear physics also provides a clear physical explanation for stability and the mathematically related phenomena of bifurcations [47] (also see Appendix A).