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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]

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.[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:

  while  
with g the gravitational acceleration and h the mean water depth. The associated kernel Kww(s) is, using the inverse Fourier transform:[4]
since cww is an even function of the wavenumber k.
   
with δ(s) the Dirac delta function.
  and     with  
The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:[6]
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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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.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, Bibcode:1978RSPTA.289..373F, CiteSeerX 10.1.1.67.6331, doi:10.1098/rsta.1978.0064, S2CID 7333207
  • 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 119121867
  • 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 55302388
  • 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 122802187
  • Whitham, G.B. (1974), Linear and nonlinear waves, Wiley-Interscience, doi:10.1002/9781118032954, ISBN 978-0-471-94090-6