Whitham equation

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:

${\displaystyle {\frac {\partial \eta }{\partial t}}+\alpha \eta {\frac {\partial \eta }{\partial x}}+\int _{-\infty }^{+\infty }K(x-\xi )\,{\frac {\partial \eta (\xi ,t)}{\partial \xi }}\,{\text{d}}\xi =0.}$

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

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]

${\displaystyle c_{\text{ww}}(k)={\sqrt {{\frac {g}{k}}\,\tanh(kh)}},}$  while  ${\displaystyle \alpha _{\text{ww}}={\frac {3}{2}}{\sqrt {\frac {g}{h}}},}$

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

${\displaystyle K_{\text{ww}}(s)={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }c_{\text{ww}}(k)\,{\text{e}}^{iks}\,{\text{d}}k={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }c_{\text{ww}}(k)\,\cos(ks)\,{\text{d}}k,}$

since c_ww 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 c_ww(k) for long waves with kh ≪ 1: ^[4]

${\displaystyle c_{\text{kdv}}(k)={\sqrt {gh}}\left(1-{\frac {1}{6}}k^{2}h^{2}\right),}$ ${\displaystyle K_{\text{kdv}}(s)={\sqrt {gh}}\left(\delta (s)+{\frac {1}{6}}h^{2}\,\delta ^{\prime \prime }(s)\right),}$ ${\displaystyle \alpha _{\text{kdv}}={\frac {3}{2}}{\sqrt {\frac {g}{h}}},}$

with δ(s) the Dirac delta function.

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

${\displaystyle K_{\text{fw}}(s)={\frac {1}{2}}\nu {\text{e}}^{-\nu |s|}}$  and  ${\displaystyle c_{\text{fw}}={\frac {\nu ^{2}}{\nu ^{2}+k^{2}}},}$  with  ${\displaystyle \alpha _{\text{fw}}={\frac {3}{2}}.}$

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

${\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}-\nu ^{2}\right)\left({\frac {\partial \eta }{\partial t}}+{\frac {3}{2}}\,\eta \,{\frac {\partial \eta }{\partial x}}\right)+{\frac {\partial \eta }{\partial x}}=0.}$

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

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

• 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, V.M. (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