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

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• 作者：Brenier ; Yann ; Levy ; Doron
• 关键词：05.45.&minus ; a ; 02.70.Bf ; KdV-type equations ; Finite-difference methods ; Nonlinear dynamics
• 刊名：Physica D
• 出版年：2000
• 期刊代码：55_01672789
• 类别：ph
• 出版时间：March 15, 2000
• 卷：137
• 期：3-4
• 页码：277-294
• 文件大小：399 K

摘要

The KdV equation can be considered as a special case of the general equation u_t+f(u)_x−δg(u_xx)_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.