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, \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 (see, e.g., [whitham], [lax-levermore], [drazin] and the references therein). 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 $\delta$ tends to zero, the solutions strongly converge to the unique entropy solution of the formal limit equation, in total contrast with the solutions of the KdV equation.