Non-local model for non-linear dispersive waves
In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves. 
The equation is notated as follows:
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This integro-differential equation for the oscillatory variable "η"("x","t") is named after Gerald Whitham who introduced it as a model to study breaking of non-linear dispersive water waves in 1967. Wave breaking – bounded solutions with unbounded derivatives – for the Whitham equation has recently been proven. 
For a certain choice of the kernel "K"("x" − "ξ") it becomes the Fornberg–Whitham equation.
Water waves.
Using the Fourier transform (and its inverse), with respect to the space coordinate "x" and in terms of the wavenumber "k":
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with "g" the gravitational acceleration and "h" the mean water depth. The associated kernel "K"ww("s") is, using the inverse Fourier transform:
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since "c"ww is an even function of the wavenumber "k".
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with "δ"("s") the Dirac delta function.
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The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:
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This equation is shown to allow for peakon solutions – as a model for waves of limiting height – as well as the occurrence of wave breaking (shock waves, absent in e.g. solutions of the Korteweg–de Vries equation).
Notes and references.
Notes.
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References.
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