摘要
The KdV equation can be considered as a special case of the general equation ut+f(u)x−δg(uxx)x=0, δ>0, where f is non-linear and g is linear, namely f(u)=u2/2 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., [P.G. Drazin, Solitons, London Math. Soc. Lect. Note Ser. 85, Cambridge University Press, Cambridge, 1983; P.D. Lax, C.D. Levermore, The small dispersion limit of the Korteweg–de Vries equation, III, Commun. Pure Appl. Math. 36 (1983) 809–829; G.B. Whitham, Linear and Nonlinear Waves, Wiley/Interscience, New York, 1974] 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.