We consider a Whitham equation as an alternative for the Korteweg–de Vries (KdV) equation in which the third derivative is replaced by the integral of a kernel, i.e., η_xxx in the KdV equation is replaced by ${\int }_{-\infty }^{\infty }{K}_{\nu }(x-\xi ){\eta }_{\xi }(\xi ,t)d\xi$. The kernel K_ν(x) satisfies the conditions lim_ν→∞K_ν(x) = δ″(x), where δ(x) is the Dirac delta function and lim_ν→0K_ν(x) = 0. The questions studied here, by means of numerical examples, are whether adjustment of the parameter ν produces both continuous solutions and shocks of the kernel equation and how well they represent KdV solutions and solutions of the underlying hyperbolic system. A typical example is for resonant forced oscillations in a closed shallow water tank governed by the kernel equation, which are compared with those governed by a partial differential equation. The continuous solutions of the kernel equation associated with frequency dispersion in the KdV equations limit to the shocks of the shallow water equations as ν → 0. Two experimental problems are solved in a single equation. As another example, suitable adjustment of ν in the kernel equation produces solutions reminiscent of a hydraulic and undular bore.

中文翻译：

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包含破裂波和连续解的Whitham方程的数值研究

我们考虑一个惠瑟姆方程作为Korteweg-DE，其中第三衍生物由内核的积分代替弗里斯（KDV）方程，即，替代η _XXX的KdV方程中被替换${\int }_{-\infty }^{\infty }{ķ}_{\nu }（X-\xi ）{\eta }_{\xi }（\xi ，Ť）d\xi$。内核ķ _ν（X）满足条件LIM _{ν →交通∞} ķ _ν（X）= δ “（X），其中，δ（X）是狄拉克δ函数和Lim _{ν →交通0} ķ _ν（X）= 0。此处通过数值示例研究的问题是是否调整参数ν产生核方程的连续解和冲击，以及它们如何很好地表示KdV解和基础双曲线系统的解。一个典型的例子是在封闭的浅水箱中由核方程控制的共振强迫振荡，并将其与由偏微分方程控制的强迫振荡进行比较。KdV方程中与频率离散相关的核方程的连续解将ν →0限制在浅水方程的激波中。在一个方程中解决了两个实验问题。作为另一个示例，在内核方程式中对ν进行适当的调整会产生使人联想到液压孔和波浪孔的解决方案。