  We examine the effect of dissipation on traveling waves in nonlinear
dispersive systems modeled by Benjamin- Bona- Mahony (BBM)-like equations. In
the absence of dissipation the BBM-like equations are found to support soliton
and compacton/anticompacton solutions depending on whether the dispersive term
is linear or nonlinear. We study the influence of increasing nonlinearity of
the medium on the soliton- and compacton dynamics. The dissipative effect is
found to convert the solitons either to undular bores or to shock-like waves
depending on the degree of nonlinearity of the equations. The anticompacton
solutions are also transformed to undular bores by the effect of dissipation.
But the compactons tend to vanish due to viscous effects. The local oscillatory
structures behind the bores and/or shock-like waves in the case of solitons and
anticompactons are found to depend sensitively both on the coefficient of
viscosity and solution of the unperturbed problem.
