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

D. Levy·Y. Brenier
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Abstract

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, \qquad \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 . 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 $\delta -> 0$ the solutions converge to the unique entropy solution of the formal limit equation, in total contrast with the solutions of the KdV equation.

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D. Levy, Y. Brenier. 1999-11-13. Dissipative Behavior of Some Fully Non-Linear KdV-Type Equations. https://arxiv.org/abs/math-ph/9911019

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