Whitham equation

Short description: 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+αη∂η∂x+∫−∞+∞K(x−ξ)∂η(ξ,t)∂ξdξ=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.

## Contents

* 1 Water waves
* 2 Notes and references
* 2.1 Notes
* 2.2 References

## 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]

cww(k)=gktanh⁡(kh), while  αww=32gh,

with g the gravitational acceleration and h the mean water depth. The associated kernel Kww(s) is, using the inverse Fourier transform:[4]

Kww(s)=12π∫−∞+∞cww(k)eiksdk=12π∫−∞+∞cww(k)cos⁡(ks)dk,
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]

ckdv(k)=gh(1−16k2h2), Kkdv(s)=gh(δ(s)+16h2δ′′(s)), αkdv=32gh,

with δ(s) the Dirac delta function.

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

Kfw(s)=12νe−ν|s| and  cfw=ν2ν2+k2, with  αfw=32.

The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:[6]

(∂2∂x2−ν2)(∂η∂t+32η∂η∂x)+∂η∂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

1. ↑ (Debnath 2005)
2. ↑ (Naumkin Shishmarev)
3. ↑ 3.0 3.1 (Whitham 1974)
4. ↑ 4.0 4.1 4.2 4.3 (Whitham 1967)
5. ↑ (Hur 2017)
6. ↑ 6.0 6.1 6.2 (Fornberg Whitham)

### References

* 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.a4
* 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, doi:10.1098/rsta.1978.0064, Bibcode: 1978RSPTA.289..373F
* Hur, Vera Mikyoung (2017), "Wave breaking in the Whitham equation", Advances in Mathematics 317: 410–437, doi:10.1016/j.aim.2017.07.006
* Moldabayev, D.; Kalisch, H.; Dutykh, D. (2015), "The Whitham Equation as a model for surface water waves", Physica D: Nonlinear Phenomena 309: 99–107, doi:10.1016/j.physd.2015.07.010, Bibcode: 2015PhyD..309...99M
* 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, doi:10.1098/rspa.1967.0119, Bibcode: 1967RSPSA.299....6W
* Whitham, G.B. (1974), Linear and nonlinear waves, Wiley-Interscience, doi:10.1002/9781118032954, ISBN 978-0-471-94090-6, Bibcode: 1974lnw..book.....W

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