We complete a full classification of nondegenerate traveling waves of scalar balance laws from
the point of view of spectral and nonlinear stability/instability under (piecewise) smooth
perturbations. A striking feature of our analysis is the elucidation of the prominent role of
characteristic points in the determination of both the spectra of the linearized operators
and the phase dynamics involved in the nonlinear large-time evolution. For a generic class of
equations an upshot of our analysis is a dramatic reduction from a tremendously wide variety
of entropic traveling waves to a relatively small range of
stable entropic traveling waves.
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