Whitham equation

In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves.

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. 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:
 * For surface gravity waves, the phase speed c(k) as a function of wavenumber k is taken as:



c_\text{ww}(k) = \sqrt{ \frac{g}{k}\, \tanh(kh)}, $$ while  $$\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, using the inverse Fourier transform:



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 cww 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 cww(k) for long waves with kh ≪ 1:



c_\text{kdv}(k) = \sqrt{gh} \left( 1 - \frac{1}{6} k^2 h^2 \right), $$ $$  K_\text{kdv}(s) = \sqrt{gh} \left( \delta(s) + \frac{1}{6} h^2\, \delta^{\prime\prime}(s) \right), $$ $$  \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 Kfw(s) – non-dimensionalised using g and h:


 * $$K_\text{fw}(s) = \frac12 \nu \text{e}^{-\nu |s|}$$ and  $$c_\text{fw} = \frac{\nu^2}{\nu^2+k^2},$$  with  $$\alpha_\text{fw}=\frac32.$$


 * The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:



\left( \frac{\partial^2}{\partial x^2} - \nu^2 \right) \left(    \frac{\partial \eta}{\partial t}    + \frac32\, \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).