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For a certain choice of the kernel K(x − ξ) it becomes the Fornberg–Whitham equation.
- with g the gravitational acceleration and h the mean water depth. The associated kernel Kww(s) is:
- The Korteweg–de Vries equation emerges when retaining the first two terms of a series expansion of cww(k) for long waves with kh ≪ 1:
- 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:
- 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
- Debnath, L. (2005), Nonlinear Partial Differential Equations for Scientists and Engineers, Springer, ISBN 9780817643232
- 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, doi:10.1098/rsta.1978.0064
- 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
- Whitham, G.B. (1974), Linear and nonlinear waves, Wiley-Interscience, ISBN 0-471-94090-9
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