一些完全非线性KdV型方程的耗散行为
Dissipative Behavior of Some Fully Non-Linear KdV-Type Equations
AI总结：
通过数值实验研究当非线性函数g为偶凹函数时，完全非线性KdV型方程在δ→0极限下的耗散行为，发现解收敛到形式极限方程的唯一熵解，与KdV方程的色散行为形成鲜明对比。
摘要：
KdV方程可视为一般方程 u_{t} + f(u)_{x} - δg(u_{xx})_x = 0, δ>0 的特例，其中 f 非线性而 g 线性，即 f(u)=u^2/2 和 g(v)=v。当参数 δ 趋于0时，KdV方程的色散行为已被深入研究。我们通过数值证据表明，当 g 为非线性时，即 g 为偶凹函数如 g(v)=-|v| 或 g(v)=-v^2，会出现完全不同的耗散行为。特别地，我们的数值结果暗示，当 δ→0 时，解收敛到形式极限方程的唯一熵解，与KdV方程的解形成鲜明对比。
英文摘要：
The KdV equation can be considered as a special case of the general equation u_{t} + f(u)_{x} - δg(u_{xx})_x = 0, \qquad δ> 0, where f is non-linear and g is linear, namely $f(u)=u^2/2$ and g(v)=v. As the parameter $δ$ 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 $δ-> 0$ 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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