Whitham equation

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

The equation is notated as follows:

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

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] Wave breaking – bounded solutions with unbounded derivatives – for the Whitham equation has recently been proven.^[5]

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:

• For surface gravity waves, the phase speed c(k) as a function of wavenumber k is taken as:^[4]

${\displaystyle c_{\text{ww}}(k)={\sqrt {{\frac {g}{k}}\,\tanh(kh)}},}$   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 K_ww(s) is, using the inverse Fourier transform:^[4]

${\displaystyle K_{\text{ww}}(s)={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }c_{\text{ww}}(k)\,{\text{e}}^{iks}\,{\text{d}}k={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }c_{\text{ww}}(k)\,\cos(ks)\,{\text{d}}k,}$

since c_ww is an even function of the wavenumber k.

• The Korteweg–de Vries equation (KdV equation) emerges when retaining the first two terms of a series expansion of c_ww(k) for long waves with kh ≪ 1:^[4]

${\displaystyle c_{\text{kdv}}(k)={\sqrt {gh}}\left(1-{\frac {1}{6}}k^{2}h^{2}\right),}$   ${\displaystyle K_{\text{kdv}}(s)={\sqrt {gh}}\left(\delta (s)+{\frac {1}{6}}h^{2}\,\delta ^{\prime \prime }(s)\right),}$   ${\displaystyle \alpha _{\text{kdv}}={\frac {3}{2}}{\sqrt {\frac {g}{h}}},}$

with δ(s) the Dirac delta function.

• Bengt Fornberg and Gerald Whitham studied the kernel K_fw(s) – non-dimensionalised using g and h:^[6]

${\displaystyle K_{\text{fw}}(s)={\frac {1}{2}}\nu {\text{e}}^{-\nu |s|}}$   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:^[6]

${\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}-\nu ^{2}\right)\left({\frac {\partial \eta }{\partial t}}+{\frac {3}{2}}\,\eta \,{\frac {\partial \eta }{\partial x}}\right)+{\frac {\partial \eta }{\partial x}}=0.}$

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).^[6][3]