Dissipative behavior of some fully non-linear KdV-type equations

The KdV equation can be considered as a special case of the general equatio n 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., [P.G. Drazin, Solitons, London Math. Sec. Lect. No te Ser. 85, Cambridge University Press, Cambridge, 1983; P.D. Lax, C.D. Lev ermore, The small dispersion limit of the Korteweg-de Vries equation, III, Commun. Pure Appl. Math. 36 (1983) 809-829; G.B. Whitham, Linear and Nonlin ear Waves, Wiley/lnterscience, New York, 1974] and the references therein). We show through numerical evidence that a completely different, dissipativ e behavior occurs when g is non-linear, namely when g is an even concave fu nction such as g(v) = -\v\ or g(v) = -v(2). In particular, our numerical re sults hint that as delta --> 0 the solutions strongly converge to the uniqu e entropy solution of the formal limit equation, in total contrast with the solutions of the KdV equation. (C) 2000 Elsevier Science B.V. All rights r eserved.