Whitham equation

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In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves:[1][2][3]

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.[4]

For a certain choice of the kernel K(x − ξ) it becomes the Fornberg–Whitham equation.

Water waves[edit]

with g the gravitational acceleration and h the mean water depth. The associated kernel Kww(s) is:[4]
with δ(s) the Dirac delta function.
  and     with  
The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:[5]
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).[5][3]

Notes and references[edit]