Title: Modulational instability in the full-dispersion Camassa-Holm equation,
Abstract: We determine the stability and instability of a sufficiently small and
periodic traveling wave to long wavelength perturbations, for a nonlinear
dispersive equation which extends a Camassa-Holm equation to include all the
dispersion of water waves and the Whitham equation to include nonlinearities of
medium amplitude waves. In the absence of the effects of surface tension, the
result qualitatively agrees with the Benjamin-Feir instability of a Stokes
wave. In the presence of the effects of surface tension, it qualitatively
agrees with those from formal asymptotic expansions of the physical problem and
it improves upon that for the Whitham equation, correctly predicting the limit
of strong surface tension. We discuss the modulational stability and
instability in the Camassa-Holm equation and related models.