Dissipative Behavior of Some Fully Non-Linear KdV-Type Equations

The KdV equation can be considered as a special case of the general equation ut + f(u)x − δg(uxx)x = 0, δ > 0, (0.1) where f is non-linear and g is linear, namely f(u) = u/2 and g(v) = v. As the parameter δ tends to 0, the dispersive behavior of the KdV equation has been throughly investigated (see, e.g., [11], [6], [2] 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) = −v2. In particular, our numerical results hint that as δ → 0 the solutions strongly converge to the unique entropy solution of the formal limit equation, in total contrast with the solutions of the KdV equation. PACS. 05.45.-a; 02.70.Bf.