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:

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]

Notes and references

Notes

References