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]

  \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[edit]

  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:[4]

  K_\text{ww}(s) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} c_\text{ww}(k)\, \text{e}^{iks}\, \text{d}k.
  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.
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:[5]

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

Notes and references[edit]