Whitham equation

Non-local model for non-linear dispersive waves

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:

∂ 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.[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

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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]

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:[4]

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.

* 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:[4]

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.

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

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:[6]

( ∂ 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).[6][3]

## Notes and references

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### Notes

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

### References

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* Debnath, L. (2005), Nonlinear Partial Differential Equations for Scientists and Engineers, Springer, ISBN 9780817643232
* Fetecau, R.; Levy, Doron (2005), "Approximate Model Equations for Water Waves", Communications in Mathematical Sciences, 3 (2): 159–170, doi:10.4310/CMS.2005.v3.n2.a4159-170&rft.date=2005&rft_id=info:doi/10.4310/CMS.2005.v3.n2.a4&rft.aulast=Fetecau&rft.aufirst=R.&rft.au=Levy,+Doron&rfr_id=info:sid/en.wikipedia.org:Whitham+equation" class="Z3988">
* Fornberg, B.; Whitham, G.B. (1978), "A Numerical and Theoretical Study of Certain Nonlinear Wave Phenomena", Philosophical Transactions of the Royal Society A, 289 (1361): 373–404, Bibcode:1978RSPTA.289..373F, CiteSeerX 10.1.1.67.6331, doi:10.1098/rsta.1978.0064, S2CID 7333207373-404&rft.date=1978&rft_id=https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.67.6331#id-name=CiteSeerX&rft_id=https://api.semanticscholar.org/CorpusID:7333207#id-name=S2CID&rft_id=info:doi/10.1098/rsta.1978.0064&rft_id=info:bibcode/1978RSPTA.289..373F&rft.aulast=Fornberg&rft.aufirst=B.&rft.au=Whitham,+G.B.&rfr_id=info:sid/en.wikipedia.org:Whitham+equation" class="Z3988">
* Hur, Vera Mikyoung (2017), "Wave breaking in the Whitham equation", Advances in Mathematics, 317: 410–437, arXiv:1506.04075, doi:10.1016/j.aim.2017.07.006, S2CID 119121867410-437&rft.date=2017&rft_id=info:arxiv/1506.04075&rft_id=https://api.semanticscholar.org/CorpusID:119121867#id-name=S2CID&rft_id=info:doi/10.1016/j.aim.2017.07.006&rft.aulast=Hur&rft.aufirst=Vera+Mikyoung&rfr_id=info:sid/en.wikipedia.org:Whitham+equation" class="Z3988">
* Moldabayev, D.; Kalisch, H.; Dutykh, D. (2015), "The Whitham Equation as a model for surface water waves", Physica D: Nonlinear Phenomena, 309: 99–107, arXiv:1410.8299, Bibcode:2015PhyD..309...99M, doi:10.1016/j.physd.2015.07.010, S2CID 5530238899-107&rft.date=2015&rft_id=info:arxiv/1410.8299&rft_id=https://api.semanticscholar.org/CorpusID:55302388#id-name=S2CID&rft_id=info:doi/10.1016/j.physd.2015.07.010&rft_id=info:bibcode/2015PhyD..309...99M&rft.aulast=Moldabayev&rft.aufirst=D.&rft.au=Kalisch,+H.&rft.au=Dutykh,+D.&rfr_id=info:sid/en.wikipedia.org:Whitham+equation" class="Z3988">
* Naumkin, P.I.; Shishmarev, I.A. (1994), Nonlinear Nonlocal Equations in the Theory of Waves, American Mathematical Society, ISBN 9780821845738
* Whitham, G.B. (1967), "Variational methods and applications to water waves", Proceedings of the Royal Society A, 299 (1456): 6–25, Bibcode:1967RSPSA.299....6W, doi:10.1098/rspa.1967.0119, S2CID 1228021876-25&rft.date=1967&rft_id=https://api.semanticscholar.org/CorpusID:122802187#id-name=S2CID&rft_id=info:doi/10.1098/rspa.1967.0119&rft_id=info:bibcode/1967RSPSA.299....6W&rft.aulast=Whitham&rft.aufirst=G.B.&rfr_id=info:sid/en.wikipedia.org:Whitham+equation" class="Z3988">
* Whitham, G.B. (1974), Linear and nonlinear waves, Wiley-Interscience, Bibcode:1974lnw..book.....W, doi:10.1002/9781118032954, ISBN 978-0-471-94090-6