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Whitham equation

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Non-local model for non-linear dispersive waves

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

The equation is notated as follows:

∂ η ∂ t + α η ∂ η ∂ x + ∫ − ∞ + ∞ K ( x − ξ ) ∂ η ( ξ , t ) ∂ ξ d ξ = 0. {\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. 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:

c ww ( k ) = g k tanh ⁡ ( k h ) , {\displaystyle c_{\text{ww}}(k)={\sqrt {{\frac {g}{k}}\,\tanh(kh)}},}   while   α ww = 3 2 g h , {\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, using the inverse Fourier transform:
K ww ( s ) = 1 2 π ∫ − ∞ + ∞ c ww ( k ) e i k s d k = 1 2 π ∫ − ∞ + ∞ c ww ( k ) cos ⁡ ( k s ) d k , {\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 cww is an even function of the wavenumber k.
c kdv ( k ) = g h ( 1 − 1 6 k 2 h 2 ) , {\displaystyle c_{\text{kdv}}(k)={\sqrt {gh}}\left(1-{\frac {1}{6}}k^{2}h^{2}\right),}   K kdv ( s ) = g h ( δ ( s ) + 1 6 h 2 δ ′ ′ ( s ) ) , {\displaystyle K_{\text{kdv}}(s)={\sqrt {gh}}\left(\delta (s)+{\frac {1}{6}}h^{2}\,\delta ^{\prime \prime }(s)\right),}   α kdv = 3 2 g h , {\displaystyle \alpha _{\text{kdv}}={\frac {3}{2}}{\sqrt {\frac {g}{h}}},}
with δ(s) the Dirac delta function.
K fw ( s ) = 1 2 ν e − ν | s | {\displaystyle K_{\text{fw}}(s)={\frac {1}{2}}\nu {\text{e}}^{-\nu |s|}}   and   c fw = ν 2 ν 2 + k 2 , {\displaystyle c_{\text{fw}}={\frac {\nu ^{2}}{\nu ^{2}+k^{2}}},}   with   α fw = 3 2 . {\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:
( ∂ 2 ∂ x 2 − ν 2 ) ( ∂ η ∂ t + 3 2 η ∂ η ∂ x ) + ∂ η ∂ x = 0. {\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).

Notes and references

Notes

  1. Debnath (2005, p. 364)
  2. Naumkin & Shishmarev (1994, p. 1)
  3. ^ Whitham (1974, pp. 476–482)
  4. ^ Whitham (1967)
  5. Hur (2017)
  6. ^ Fornberg & Whitham (1978)

References

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