Numerical study of a Whitham equation exhibiting both breaking waves and continuous solutions

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., ηxxx in the KdV equation is replaced by ∫−∞∞Kν(x−ξ)ηξ(ξ,t)dξ. The kernel Kν(x) satisfies the conditions limν→∞Kν(x) = δ″(x), where δ(x) is the Dirac delta function and limν→0Kν(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.