  In this paper we derive generalized forms of the Camassa-Holm (CH) equation
from a Boussinesq-type equation using a two-parameter asymptotic expansion
based on two small parameters characterizing nonlinear and dispersive effects
and strictly following the arguments in the asymptotic derivation of the
classical CH equation. The resulting equations generalize the CH equation in
two different ways. The first generalization replaces the quadratic
nonlinearity of the CH equation with a general power-type nonlinearity while
the second one replaces the dispersive terms of the CH equation with
fractional-type dispersive terms. In the absence of both higher-order
nonlinearities and fractional-type dispersive effects, the generalized
equations derived reduce to the classical CH equation that describes
unidirectional propagation of shallow water waves. The generalized equations
obtained are compared to similar equations available in the literature, and
this leads to the observation that the present equations have not appeared in
the literature.
