  We develop a direct method for solving a modified Camassa-Holm equation with
cubic nonlinearity and linear dispersion under the rapidly decreasing vanishing
boundary condition. We obtain a compact parametric representation for the
multisoliton solutions and investigate their properties. We show that the
introduction of a linear dispersive term exhibits various new features in the
structure of solutions. In particular, we find the smooth solitons whose
characteristics are different from those of the Camassa-Holm equation, as well
as the novel types of singular solitons. A remarkable feature of the soliton
solutions is that the underlying structure of the associated tau-functions is
the same as that of a model equation for shallow-water waves introduced by
Ablowitz {\it et al}(1974 {\it Stud. Appl. Math.} {\bf 53} 249-315). Finally,
we demonstrate that the short-wave limit of the soliton solutions recovers the
soliton solutions of the short pulse equation which describes the propagation
of ultra-short optical pulses in nonlinear media.
