  In recent work, we have proposed a theory for the derivation of an exact
nonlinear dispersion relation for elastic wave propagation which here we
consider for a thin rod (linearly nondispersive) and a thick rod (linearly
dispersive). The derived relation has been validated by direct time-domain
simulations examining instantaneous dispersion. In this Letter, we present a
major extension of the theory revealing that our derived nonlinear dispersion
relation provides direct and exact prediction of harmonic generation, thus
merging two key tenets of wave propagation. This fundamental unification of
nonlinear dispersion and harmonic generation is applicable to any arbitrary
wave profile characterized initially by an amplitude and a wavenumber,
irrespective of the type and strength of the nonlinearity, and regardless of
whether the medium is dispersive or nondispersive in the linear limit. Another
direct outcome of the theory is an analytical condition for soliton synthesis
based on balance between hardening and softening dispersion.
