# Whitham equation

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

${\displaystyle {\frac {\partial \eta }{\partial t}}+\alpha \eta {\frac {\partial \eta }{\partial x}}+\int _{-\infty }^{+\infty }K(x-\xi )\,{\frac {\partial \eta (\xi ,t)}{\partial \xi }}\,{\text{d}}\xi =0.}$

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

${\displaystyle c_{\text{ww}}(k)={\sqrt {{\frac {g}{k}}\,\tanh(kh)}},}$   while   ${\displaystyle \alpha _{\text{ww}}={\frac {3}{2}}{\sqrt {\frac {g}{h}}},}$
with g the gravitational acceleration and h the mean water depth. The associated kernel Kww(s) is:[4]
${\displaystyle K_{\text{ww}}(s)={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }c_{\text{ww}}(k)\,{\text{e}}^{iks}\,{\text{d}}k.}$
${\displaystyle c_{\text{kdv}}(k)={\sqrt {gh}}\left(1-{\frac {1}{6}}k^{2}h^{2}\right),}$   ${\displaystyle K_{\text{kdv}}(s)={\sqrt {gh}}\left(\delta (s)+{\frac {1}{6}}h^{2}\,\delta ^{\prime \prime }(s)\right),}$   ${\displaystyle \alpha _{\text{kdv}}={\frac {3}{2}}{\sqrt {\frac {g}{h}}},}$
with δ(s) the Dirac delta function.
${\displaystyle K_{\text{fw}}(s)={\frac {1}{2}}\nu {\text{e}}^{-\nu s}}$   and   ${\displaystyle c_{\text{fw}}={\frac {\nu ^{2}}{\nu ^{2}+k^{2}}},}$   with   ${\displaystyle \alpha _{\text{fw}}={\frac {3}{2}}.}$
The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:[5]
${\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}-\nu ^{2}\right)\left({\frac {\partial \eta }{\partial t}}+{\frac {3}{2}}\,\eta \,{\frac {\partial \eta }{\partial x}}\right)+{\frac {\partial \eta }{\partial x}}=0.}$
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]