  We introduce a fairly general dispersive-dissipative nonlinear equation,
which is characterized by fractional Laplacian operators in both the dispersive
and dissipative terms. This equation includes some physically relevant models
of fluid dynamics as particular cases. Among them are the \emph{dispersive
Kuramoto-Velarde}, the \emph{Kuramoto-Sivashinsky} equation, and some nonlocal
perturbations of the \emph{KdV} and the \emph{Benjamin-Ono} equations. We
thoroughly study the effects of the fractional Laplacian operators in the
qualitative study of solutions: on the one hand, we prove a sharp
well-posedness result in the framework of Sobolev spaces of negative order, and
on the other hand, we investigate the pointwise decaying properties of
solutions in the spatial variable, which are optimal in some cases. These last
results are of particular interest for the corresponding physical models.
Precisely, they align with previous numerical works on the spatial decay of a
particular kind of solutions, commonly referred to as solitary waves.
