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

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Mortell, Michael P.
Mulchrone, Kieran F.
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American Institute of Physics
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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.
Integral transforms , Soliton solutions , Surface waves , Fourier analysis , Oscillating flow , Korteweg-de Vries equation , Generalized functions , Navier Stokes equations
Mortell, M. P. and Mulchrone, K. F. (2021) 'Numerical study of a Whitham equation exhibiting both breaking waves and continuous solutions', AIP Advances, 11 (4), 045002 (13 pp). doi: 10.1063/5.0047582