38
R. Grimshaw and C. Yuan
Here the notation C 4 , C 6 denote , respectively, and like b = −C 2 =
− 2
> depend on the modulus m only.
To obtain the modulation equations for a solitary wave train, we take the limit
m → 1 and then b ∼ −1∕K(m), C 4 ∼ 2∕3K(m), and C 6 ∼ 8∕15K(m). To leading
order M ∼ d
2
∕2 and P ∼ 𝛼d
3
∕3 and then both equations (31, 32) reduce to the same
equation for d alone,
d 𝜏 + 𝛼dd X = 0 ,
(35)
and so d can be regarded as a known quantity. At the same time, the cnoidal wave
expression (28) reduces to
U = a sech
2
(𝛾𝜃; m)} + d , 𝜃 = k(X − V𝜏) , V − 𝛼d =
𝛼a
3
= 12𝛾
2 k
2
, (36)
with two parameters still to be determined. The equation for conservation of waves
(30) provides one equation for k and the second equation is
{
a 2
k𝛾
} 𝜏 + V{
a 2
k𝛾
} X +
a 2
k𝛾
𝛼d X = 0 ,
(37)
This can be obtained by a more careful consideration of the limit m → 1 in the modulation equations (31, 32) by retaining the terms in 1∕K(m), or more directly by
averaging the wave action conservation law (16) directly for a solitary wave, see [7]
and the discussion in [5].
References
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Philadelphia: SIAM.
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Fluid Mechanics, 25, 241–270.
3. Benney, D. J. (1966). Long non-linear waves in fluid flows. Journal of Mathematical Physics,
45, 52–63.
4. El, G. (2007). Kortweg-de Vries equation and undular bores. In R. Grimshaw (Ed.), Solitary
waves in fluids. Advances in Fluid Mechanics (Vol. 47, pp. 19–53). WIT Press.
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flows (pp. 1–27). Boston: Kluwer.
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Angewandte Mathematik, 30, 96–109.