  The KdV equation can be considered as a special case of the general equation
u_{t} + f(u)_{x} - \delta g(u_{xx})_x = 0, \qquad \delta > 0, where f is
non-linear and g is linear, namely $f(u)=u^2/2$ and g(v)=v. As the parameter
$\delta$ tends to 0, the dispersive behavior of the KdV equation has been
throughly investigated . We show through numerical evidence that a completely
different, dissipative behavior occurs when g is non-linear, namely when g is
an even concave function such as $g(v)=-|v|$ or $g(v)=-v^2$. In particular, our
numerical results hint that as $\delta -> 0$ the solutions converge to the
unique entropy solution of the formal limit equation, in total contrast with
the solutions of the KdV equation.
